# Article:Comparing the uncomventional Marshall Pitching Motion to a Refined Conventional Pitching Motion That incorporates Linear and Angular Applied Techniques

*12,191*pages on

this wiki

Comparing the Unconventional Marshall Pitching Motion to a

Refined Conventional Pitching Motion That Incorporates

Linear and Angular Applied Techniques

By: John D’Acquisto, Ph.D

Introduction

This report compares and contrasts two different baseball pitching motions as related to

their athletic fitness levels. To clarify, these pitching motions will henceforth be referred

to as the "Unconventional Marshall Motion" and "Refined Conventional Motion." As

demonstrated by the case subject, biomechanical hallmarks of each motion and their

applied techniques will be detailed in later sections.

Scientific biomechanical study of baseball pitching poses some significant challenges.

The fastest human athletic activity known, elite level pitching arm accelerations take

place in less than two hundredths of a second. Recording body segments moving at these

speeds requires equipment that operates at micro-second levels...tools like high speed

video, high speed film and, more recently, motion sensors.

Although no two athletes perform the baseball pitching motion in exactly the same way

there are commonalities in deliveries. Researching a more efficient and anatomically

sound way to throw is difficult because the sample size is always one: one pitcher. This

study is unique because it is the focus of a professional caliber athlete who has learned

two distinct ways to throw.

The study subject is a 23 year-old male, 6'-4, 215-pound, left-handed pitcher, previously

drafted by a Major League Baseball club in 2003. From June 2004 to May 2007 the

subject participated in three years of intensive physical training while learning an

alternate mechanic under the direct guidance and supervision of Dr. Mike Marshall. In

the eighteen months prior to this study the subject reports experiencing significant pain in

the left anterior shoulder. Medical evaluation revealed signs of partial shoulder

dislocation that correlates with the subject’s training experience. Portions of this study

were designed specifically to identify stressors in the Unconventional Marshall Pitching

Motion that might have contributed to anterior shoulder pain.

Methodology & Scientific Assumptions

When analyzing the pitching motion and assessing the value of a model for the perfect

pitching motion there are individual-specific issues that need to be assessed for each

pitcher; they are like fingerprints and each is unique. Due to variances of an individual’s

skeletal and muscular structure his pitching technique bears a unique biomechanical

signature. It is suggested that assessments be made on each individual, focusing on the

overall stress conduit relative to their physical ability to perform, repeat, and maintain the

specific movements of their technique. To say that there is a perfect model for pitching is

a misnomer and should be viewed as such.

There have been cases of longevity among pitchers and their pitching motions over the

years. Most of the case subjects have physically trained their bodies to accept specific

amounts of stress at certain points of the kinetic chain sequence. The repetitive

movements specific to their motion are very consistent and maintain that consistency

throughout their career, thus decreasing the chance of mistake or injury. In this study we

are examining the consistencies, durability potential, stress factors, kinetic chain

sequences and overall injury parameters that would present potential injury or decreased

longevity to the subject athlete, as compared between the Refined Conventional Motion

and the Unconventional Marshall Pitching Motion.

The study collected objective data utilizing 3D-motion tracking sensors strategically

placed on key body segments to extract biomechanical information necessary for

comparative analysis. This analysis was not designed to provide an indication of the

subject’s ability to perform a given task (though some insights about inherent ability were

gained). Rather it is a measure of the relative efficiency of the ability of the two motions

to efficiently transfer energy from one body segment to the other and the fitness levels

needed to perform each motion as demonstrated by the subject. All objective data was

obtained using the E-Factor motion capture system developed by JZZ Technologies, Inc.,

systematized in part through analysis of elite athletes using data gathered from nine years

of studies.

Fitness levels (muscle strength relative to specific actions or motions) and relative ideals

used in the study were derived from anatomically correct kinetic chaining sequences.

Ideal sequencing was determined in part through longevity of the subjects from past

studies and from existing Torque Stand studies involving over 48,000 test cases. Using

integrated system software, fitness levels were evaluated on the basis of the sum of

muscle torques developed by main muscle groups under static conditions (ISI - isometric

strength indicator). Measurements were based in part on previous studies using an

isometric muscle torque stand (local make), which enabled the direct measuring of

torques for flexors and extensors of elbow, shoulder, knee and hip joints and flexors and

extensors of trunk. Angle positions for all joints were 90 deg (with 180 deg meaning full

extension) with the exception of shoulder joint (45 deg). The stand enabled each group of

muscles to be measured with simultaneous elimination of the influence of any other

forces on the result [Jaszczuk et al.1987].

Kinematic anatomical sequencing was examined by recording and comparing Maximum

Rotational Speeds and Progressive Speed Gains for each major body segment, measured

in degrees per second. Directional speeds in the X, Y, and Z planes were measured. Hand

speeds, relative tempo, posture at stance, posture at balance point, posture at hand break,

posture at toe touch, posture at delivery and posture at finish were all derived from the

data. The biomechanical structures were analyzed and then mathematically assessed to

determine the efficiency of the two specific pitching motions.

The diagram below shows the coordinate system used by the E-Factor analysis software

used in this study. The axes shown are used to determine spatial position as well as

orientation. All movements are recorded using sensors placed on the body that are

constantly monitored by an electromagnetic global reference frame surrounding the

athlete.

Comparative Accuracy & Efficiencies Summary

Data obtained from this study lead to the conclusion that the Refined Conventional

Motion is: 1) more accurate, and 2) more efficient, when compared to the

Unconventional Marshall Motion.

As demonstrated by the subject it was found that considerably lower hand speed

efficiencies were generated in the Unconventional Marshall Motion as compared to the

Refined Conventional Motion. Based on data compiled for both accuracy and efficiency

the subject’s hand speed generated by the Unconventional Marshall Motion was 25.4 %

and the hand speed generated by the Refined Conventional Motion was 58.3%.

Accuracy in the pitching motion references the proportional number of times that the

pitcher is likely to deliver pitches to the strike zone. Objectively comparing the Refined

Conventional Motion to the Unconventional Marshall Motion it was found that the test

subject was more likely to be inconsistent in delivering pitches to the strike zone when

utilizing the Unconventional Marshall Motion based on inconsistencies in delivery

repeatability as measured by relative efficiencies and variations in driveline

displacement.

Prior to release, using the Unconventional Marshall Motion, a significant disconnect of

the sequential energy transfer was noted, disrupting the kinematic sequencing from the

arm to the wrist. Also noted is a major 7.5 inch shift of the Center of Gravity to the right

side lead leg position. This shift (along with the arm lagging significantly behind torso

rotation) places significant stress on the shoulder capsule and was noted to

correspondingly decelerate the arm, wrist and hand. Ultimately this would result in lower

velocity of the baseball.

At the point of ball release, using the Unconventional Marshall Motion, the subject’s

Pelvis Rotation was 119.93 degrees per second and the Head Side Bend was -27.21

degrees per second and the overall Upper Body Bend was -48.61 degrees per second

indicating significant lateral displacement of the body before, during, and after release.

This inefficiency correlates with lowered release velocities as torsion force is directed

laterally away from the target rather than towards it.

Unconventional Marshall Motion: Analysis Summary

Date of Test: 14 January 2008

Location: Mtn. Pointe

Three-dimensional motion tracking sensors were strategically placed on key body

segments to extract data. Breakdown of efficiency scores follows. (Higher scores

represent a more biomechanically efficient motion.)

Efficiency Summary

Hand

Speed

Temp

o

Posture

at

Stance

Posture at

Balance

Point

Posture at

Hand

Break

Posture

at

Toe

Touch

Posture

at

Delivery

Postur

e at

Finish

25.4% 69.3% 64.2% 55.1% 53.3% 51.2% 26.8% 31.8%

Unconventional Marshall Motion: Rotational Speeds and Progressive Speed Gains

The graph and charts in this section depict the relative speeds and sequencing of the

subject’s left side joints, prior to and at ball release. Efficient ballistic athletic events will

follow a sequential delivery and forward passage of peak joint linear velocities from the

ground up through the wrist and hand.

The graph below represents the trial kinematic sequence from address to finish. The red

line represents rotational speed of the pelvis. The green line represents rotational speed of

the upper body. The blue line represents rotational speed of the arm. The yellow line

represents rotational speed of the hand.

Max Rotational Speed (degrees/sec)

Segment Subject Ideal

Pelvis 570.27 500.00

Upper Body 822.90 850.00

Arm 5197.09 5000.00

Hand 3248.23 7000.00

Progressive Speed Gains (degrees/sec)

Segment Subject Ideal

Pelvis to UBody 570.27 350.00

UBody to Arm 822.90 4150.00

Arm to Hand 5197.09 2000.00

Kinematic Sequence

Ideal Kinematic Sequence: Pelvis, Torso, Arm, Hand

Subject Kinematic Sequence: Pelvis, Torso, Hand, Arm

It is notable that the subject peaks his linear hip, shoulder, elbow and wrist velocities in

proper sequence but there is a significant drop in peak velocities at the elbow and wrist

joints approximately 0.4 seconds before ball release.

Each body segment in the kinematic sequence must transfer energy efficiently to the next

segment. Power created, beginning with the feet, is transferred through the legs, pelvis,

torso and arms to be delivered at the hand or sports implement. Each segment adds its

distinct energy increase to total power.

It is important that energy increases transfer smoothly and in proper sequence. Correct

sequencing allows for a biomechanically fluid motion that contributes more power,

explosiveness and efficiency while reducing workload and risk of injury. Abrupt gains or

losses from one segment to the next can be indicative of potential problem areas or

specific injury risks.

Unconventional Marshall Motion: Hand Speeds

The graph and chart below depicts the subject’s hand speed during the trial from stance to

finish. The red line represents the hand speed towards or away from the target (X-Axis).

The green line represents hand speed laterally across the body (Y-Axis). The blue line

represents hand speed up or down (Z-Axis). The yellow line represents overall hand

speed and is calculated based on all three vectors (X-, Y-, and Z-Axes)

Overall Hand Speed Efficiency: 25.4%

Direction Max (MPH) Min (MPH) At Delivery (MPH) Res. At Delivery (%)

X Direction 46.62 -10.05 +35.33 76.1

Y Direction 9.88 -22.75 -20.97 -45.1

Z Direction 21.15 -28.36 -21.67 -46.7

Resultant 47.02 0.07 +46.45 n/a

In the table above:

X Direction defines movement toward and away from the target. Positive values

indicate hand speed toward the target. Negative values indicate hand speed away

from the target.

Y Direction defines movement toward and away from the body laterally. Positive

values indicate lateral hand speed away from the body. Negative values indicate

lateral hand speed toward the body.

Z Direction defines movement up and down. Positive values indicate hand speed

upwards. Negative values indicated hand speed downwards.

Overall hand speed efficiency is an indicator of both accuracy and biomechanical

efficiency. The higher the score the more likely any given pitch will be delivered to the

target. Higher scores also predict less fatigue and less injury risk. The subject’s overall

hand speed efficiency of 25.4%, using the Unconventional Marshall Motion, places him

at risk for both substantial fatigue and injury and predicts that he will have difficulty in

throwing strikes with this delivery.

Unconventional Marshall Motion: Body Posture at Toe Touch

Toe touch is defined as the moment when the stride foot lands and weight has transferred

in a straight line towards the target. Simultaneously the pitcher should initiate forward

movement of the throwing arm and rotation of the pelvis and upper body. Posture at toe

touch is vital if a pitcher is to be in a strong, athletic position where upper and lower body

can work together to transfer energy efficiently.

Overall Efficiency of Body Posture At Toe Touch: 51.2%

Body Segment Subject Posture Ideal Posture Difference

Pelvis Rotation 35.60 Open 45.00 Open 9.60 Closed

Spine Rotation 19.91 Closed 15.00 Closed 4.91 Closed

Upper Body Rotation 15.63 Open 30.00 Open 14.37 Closed

Head Rotation 70.31 Open 75.00 Open 4.69 Closed

Trail Foot Rotation 84.20 Closed 35.00 Open 119.20 Closed

Lead Foot Rotation 94.93 Open 80.00 Open 14.93 Open

Pelvis Forward-Backward Bend 0.77 Forward 2.00 Forward 1.23 Backward

Spine Forward-Backward Bend 0.51 Forward 6.00 Forward 5.49 Backward

Torso Forward-Backward Bend 0.86 Forward 8.00 Forward 7.14 Backward

Head Forward-Backward Bend 3.71 Forward 10.00 Forward 6.29 Backward

Trail Foot Dorsi-Plantar Flexion 69.28 Up 25.00 Up 44.28 Up

Lead Foot Dorsi-Plantar Flexion 6.22 Up 0.00 Up 6.22 Up

Pelvis Side Bend 1.10 Lead 3.00 Lead 1.90 Trail

Spine Side Bend 9.00 Trail 5.00 Trail 4.00 Trail

Torso Side Bend 7.70 Trail 2.00 Trail 5.70 Trail

Head Side Bend 27.50 Lead 5.00 Trail 32.50 Trail

Trail Foot Pronation-Supination 5.52 Outward 20.00 Inward 25.52 Outward

Lead Foot Pronation-Supination 9.38 Inward 0.00 Outward 9.38 Inward

(Note: All measurements in degrees; Data taken from frame 743)

Unconventional Marshall Motion: Body Posture At Ball Release

Delivery or release point is defined as the instant in which the pitcher’s hand is furthest

from the body in the direction of the target while the middle finger is still in contact with

the ball. Every athlete must reach this point immediately at ball release.

Overall Efficiency of Body Posture At Ball Release: 26.8%

Body Segment Subject Posture Ideal* Posture Difference

Pelvis Rotation 122.64 Open 90.00 Open 32.64 Open

Spine Rotation 0.69 Closed 0.00 Open 0.69 Open

Upper Body Rotation 138.62 Open 90.00 Open 48.62 Open

Head Rotation 140.72 Open 90.00 Open 50.72 Open

Trail Foot Rotation 209.78 Closed 80.00 Open 289.78 Closed

Lead Foot Rotation 93.24 Open 80.00 Open 13.24 Open

Pelvis Forward-Backward Bend 15.72 Forward 10.00 Forward 5.72 Forward

Spine Forward-Backward Bend 24.50 Forward 15.00 Forward 5.72 Forward

Torso Forward-Backward Bend 38.89 Forward 25.00 Forward 13.89 Forward

Head Forward-Backward Bend 26.14 Forward 2.00 Forward 24.14 Forward

Trail Foot Dorsi-Plantar Flexion 42.16 Upward 85.00 Upward 42.84 Downward

Lead Foot Dorsi-Plantar Flexion 9.00 Upward 0.00 Upward 9.00 Upward

Pelvis Side Bend 2.43 Trail 0.00 Trail 2.43 Trail

Spine Side Bend 48.94 Lead 10.00 Trail 58.94 Trail

Torso Side Bend 44.38 Lead 10.00 Trail 54.38 Lead

Head Side Bend 26.98 Lead 0.00 Trail 27.98 Lead

Trail Foot Pronation-Supination 2.04 Outward 7.00 Inward 9.04 Outward

Lead Foot Pronation-Supination 34.52 Inward 0.00 Outward 34.52 Inward

(Note: All measurements in degrees; Data taken from frame 805)

- Note that in the subject’s graphs and charts derived from the E-Factor system the word

IDEAL is used. This is not based on a conceptualized model but rather a computation of

all of the analysis done in the past of subjects who scored high with respect to overall

fitness to their specific athletic endeavor.

It was noted that the subject’s linear movement reaches peak approximately 0.3 seconds

prior to ball release and his rotational momentum reaches peak at ball release. This

indicates a pre-release linear braking motion of the body. This action will place higher

stress on the anterior of the shoulder joint as it positions itself for energy transfer to the

elbow, wrist and hand.

The following chart and robotic representation of the subject addresses the relative

positions of shoulders vs. hips at ball release using the Unconventional Marshall Motion.

It is noted that at ball release the subject’s shoulders are only 17 degrees forward of his

hips. This indicates poor utilization of trunk rotational torque in the delivery of the pitch.

(Also see enlarged version of the robotic representation below.) '

Date of Test: 14 January 2008

Location: Mtn. Pointe

Three-dimensional motion tracking sensors were strategically placed on key body

segments to extract data. Breakdown of efficiency scores follows. (Higher scores

represent a more biomechanically efficient motion.)

Efficiency Summary

Hand

Speed

Temp

o

Posture

at

Stance

Posture at

Balance

Point

Posture at

Hand

Break

Posture

at

Toe

Touch

Posture

at

Delivery

Posture

at

Finish

58.3% 66.5% 51.9% 44.1% 44.3% 37.8% 28.1% 31.0% '

The graph and charts in this section depict the relative speeds and sequencing of the

subject’s left side joints, prior to and at ball release. Efficient ballistic athletic events will

follow a sequential delivery and forward passage of peak joint linear velocities from the

ground up through the wrist and hand.

The graph below represents the trial kinematic sequence from address to finish. The red

line represents rotational speed of the pelvis. The green line represents rotational speed of

the upper body. The blue line represents rotational speed of the arm. The yellow line

represents rotational speed of the hand.

Max Rotational Speed (degrees/sec)

Segment Subject Ideal

Pelvis 619.96 500.00

Upper Body 868.52 850.00

Arm 5905.61 5000.00

Hand 4391.86 7000.00

Progressive Speed Gains (degrees/sec)

Segment Subject Ideal

Pelvis to UBody 248.56 350.00

UBody to Arm 5037.10 4150.00

Arm to Hand -1513.95 2000.00

Kinematic Sequence

Ideal Kinematic Sequence: Pelvis, Torso, Arm, Hand

Subject Kinematic Sequence: Pelvis, Torso, Hand, Arm

Each body segment in the kinematic sequence must transfer energy efficiently to the next

segment. Power created, beginning with the feet, is transferred through the legs, pelvis,

torso and arms to be delivered at the hand or sports implement. Each segment adds its

distinct energy increase to total power.

It is important that energy increases transfer smoothly and in proper sequence. Correct

sequencing allows for a biomechanically fluid motion that contributes more power,

explosiveness and efficiency while reducing workload and risk of injury. Abrupt gains or

losses from one segment to the next can be indicative of potential problem areas or

specific injury risks.

Refined Conventional Motion: Hand Speeds

The graph below represents subject hand speed during the trial from stance to finish. The

red line represents the hand speed towards or away from the target (X-Axis). The green

line represents hand speed laterally across the body (Y-Axis). The blue line represents

hand speed up or down (Z-Axis). The yellow line represents overall hand speed and is

calculated based on all three vectors (X-, Y-, and Z-Axes)

Overall Hand Speed Efficiency: 58.3%

Direction Max (MPH) Min (MPH) At Delivery (MPH) Res. At Delivery (%)

X Direction 60.28 -18.73 47.38 97.2

Y Direction 13.14 -21.76 -10.56 -21.7

Z Direction 22.65 -31.18 -4.32 -8.9

Resultant 62.04 0.06 48.73 n/a

In the table above:

X Direction defines movement toward and away from the target. Positive values

indicate hand speed toward the target. Negative values indicate hand speed away

from the target.

Y Direction defines movement toward and away from the body laterally. Positive

values indicate lateral hand speed away from the body. Negative values indicate

lateral hand speed toward the body.

Z Direction defines movement up and down. Positive values indicate hand speed

upwards. Negative values indicated hand speed downwards.

Overall hand speed efficiency is an indicator of both accuracy and biomechanical

efficiency. The higher the score the more likely any given pitch will be delivered to the

target. Higher scores also predict less fatigue and less injury risk. The subjects overall

hand speed efficiency of 58.3%, using the Refined Conventional Motion, places him at

risk for fatigue yet lowers the risk of injury as compared to the Unconventional Marshall

Motion. '

Toe touch is defined as the moment when the stride foot lands and weight has transferred

in a straight line towards the target. Simultaneously the pitcher should initiate forward

movement of the throwing arm and rotation of the pelvis and upper body. Posture at toe

touch is vital if a pitcher is to be in a strong, athletic position where upper and lower body

can work together to transfer energy efficiently.

Overall Efficiency of Body Posture At Toe Touch: 37.8%

Body Segment Subject Posture Ideal Posture Difference

Pelvis Rotation 51.06 Open 45.00 Open 6.06 Open

Spine Rotation 33.93 Closed 15.0 Closed 18.93 Closed

Upper Body Rotation 17.01 Open 30.00 Open 12.99 Closed

Head Rotation 75.66 Open 75.00 Open 0.66 Open

Trail Foot Rotation 84.48 Open 35.00 Open 49.48 Open

Lead Foot Rotation 93.90 Open 80.00 Open 13.90 Open

Pelvis Forward-Backward Bend 2.52 Backward 2.00 Forward 4.52 Backward

Spine Forward-Backward Bend 7.10 Backward 6.00 Forward 13.10 Backward

Torso Forward-Backward Bend 7.66 Backward 8.00 Forward 15.66 Backward

Head Forward-Backward Bend 2.67 Forward 10.00 Forward 7.33 Backward

Trail Foot Dorsi-Plantar Flexion 66.68 Up 25.00 Up 41.68 Up

Lead Foot Dorsi-Plantar Flexion 7.65 Up 0.00 Up 7.65 Up

Pelvis Side Bend 2.82 Trail 3.00 Lead 5.82 Trail

Spine Side Bend 7.84 Lead 5.00 Trail 12.84 Lead

Torso Side Bend 4.09 Lead 2.00 Trail 6.09 Lead

Head Side Bend 37.71 Lead 5.00 Trail 42.71 Lead

Trail Foot Pronation-Supination 4.78 Outward 20.00 Inward 24.78 Outward

Lead Foot Pronation-Supination 18.34 Outward 0.00 Outward 18.34 Outward

(Note: All measurements in degrees; Data taken from frame 664)

Refined Conventional Motion: Body Posture At Ball Release

Delivery or release point is defined as the instant in which the pitcher’s hand is furthest

from the body in the direction of the target while the middle finger is still in contact with

the ball. Every athlete must reach this point immediately at ball release.

Overall Efficiency of Body Posture At Delivery: 28.1%

Body Segment Subject Posture Ideal Posture Difference

Pelvis Rotation 118.98 Open 90.00 Open 28.98 Open

Spine Rotation 6.83 Closed 0.00 Open 6.83 Closed

Upper Body Rotation 134.46 Open 90.00 Open 44.46 Open

Head Rotation 134.09 Open 90.00 Open 44.09 Open

Trail Foot Rotation 150.00 Open 80.00 Open 70.00 Open

Lead Foot Rotation 93.38 Open 80.00 Open 13.38 Open

Pelvis Forward-Backward Bend 21.48 Forward 10.00 Forward 11.48 Forward

Spine Forward-Backward Bend 20.31 Forward 15.00 Forward 5.31 Forward

Torso Forward-Backward Bend 40.73 Forward 25.00 Forward 15.73 Forward

Head Forward-Backward Bend 17.18 Forward 2.00 Forward 15.18 Forward

Trail Foot Dorsi-Plantar Flexion 57.13 Upward 85.00 Upward 27.87 Down

Lead Foot Dorsi-Plantar Flexion 10.48 Upward 0.00 Upward 10.48 Upward

Pelvis Side Bend 4.32 Trail 0.00 Trail 4.32 Trail

Spine Side Bend 47.97 Lead 10.00 Trail 57.97 Lead

Torso Side Bend 41.91 Lead 10.00 Trail 51.91 Lead

Head Side Bend 24.37 Lead 0.00 Trail 24.37 Lead

Trail Foot Pronation-Supination 4.61 Outward 7.00 Inward 11.61 Outward

Lead Foot Pronation-Supination 4.55 Inward 0.00 Outward 4.55 Inward

(Note: All measurements in degrees; Data taken from frame 706)

The following chart and robotic representation of the subject addresses the relative

positions of shoulders vs. hips at ball release using the Refined Conventional Motion. '

Within a reasonable degree of scientific probability, the subject’s Unconventional

Marshall Motion pitching technique was contributing to an acute overuse syndrome of

the anterior left shoulder. When using the Refined Conventional Motion it was

determined that the subject’s point of release was more consistent as well as his ability to

throw the ball to spots accurately without pain in the anterior left shoulder capsule.

The subject, after 3 days of training with the Refined Conventional Motion, was adjusting

to his new pitching style. He was pain free and pitching at 100% effort, although feeling

that he is only applying 85% of his effort.

During the Refined Conventional Motion assessment, the motion direction of player's

center of gravity is consistent with the direction of ball flight, so it has an initial velocity

before release. The results of strength assessments (both under static and dynamic

conditions) cannot be directly compared to others results because the unconventional

measurements procedure was applied.

Overall hand speed efficiency is an indicator of both accuracy and biomechanical

efficiency. The higher the score the more likely any given pitch will be delivered to the

target. Higher scores also predict less fatigue and less injury risk. The subjects overall

hand speed efficiency of 25.4%, using the Unconventional Marshall Motion, places him

at risk for both substantial fatigue and injury and predicts that he will have difficulty in

throwing strikes with this delivery. When tested using the Refined Conventional Motion,

the hand speed efficiencies were 58.3%, which is a considerable improvement from the

Marshall Pitching Motion, yet will require a specific training regimen to minimize

fatigue.

Recommendations

From a biomechanical prospective, to minimize shoulder stress in this subject’s pitching

technique, it was recommended that he strive to relax his shoulder and utilize better

control of trunk torque in the delivery of forces from the ground through the hand. He

should delay ball release somewhat until his torso is in a more forward flexed posture and

the upper arm segment is oriented more towards a parallel ground position. This will shift

the stress point away from the anterior shoulder capsule. The additional body stretch and

torque will eventually translate to higher kinetic energy and faster ball delivery.

It was stressed that changing his pitching style from the Unconventional Motion to a

more revised Conventional Motion will adversely effect his accuracy and ball delivery

ability until he re-adjusts his body coordination. It will be necessary for him to work on

total body flexibility in order to achieve the new desired ball release position.

Discussion

The subject’s basic form is good in demonstrating both motions, but there is room for

improvement in all aspects of his delivery. He is noted to have a pre-mature braking

action of forward momentum causing him to release the ball in an upright posture that

places the burden of shoulder capsule stress in the anterior compartment. This release

posture was verified by examining the relatively close positioning of shoulders vs. hips

and left shoulder vs. center of gravity.

Additionally, using the glove-side arm to generate downward planar tilt to the shoulder

line before torso rotation occurs appears to be a significant solution both in theory and in

practice. Narrowing the rotational axis of shoulder/torso rotation as viewed from

overhead provides a more linear Z vector. In theory, this could minimize the impact of

destructive centrifugal forces. Although there is some centrifugal force generated it

appears likely that an athlete can control this with a more vertical forearm through

release, initiated in part by action of the pronator teres and pronator quadratus.

Many researchers who have investigated an overhand throw have indicated that muscle

strength is a very important factor influencing throwing velocity [Pauwels 1978,

Pedegana et al.1982, Amin et al.1985, Pawlowski and Perrin 1989, Renne et al.1990,

Wooden et al.1992, Bartlet et al.1993, Eliasz 1993, Marczinka 1993]. In this work

statistical analysis has shown that the muscle strength of trunk flexors is one of the most

significant velocity determinants in analyzed throws (this variable is in all presented

equations). Abdominal muscles (rectus abdominis, external and internal oblique muscles)

serve as the primary trunk flexors. These muscles, acting together, are involved in

forward bending but trunk rotation is caused by one-side shortening action of external

and internal oblique muscles. Both types of motions can be observed during throwing,

before release [Atwater 1980, Joris et al.1985, Eliasz 1993, Marczinka 1993].

The investigation has some practical applications. There are two significant possibilities

to improve throwing velocity, likely in all pitching techniques: (1) by developing strength

of specific abdominal muscles, and (2) by improving speed of external and internal

rotation at shoulder joint. [Joris et al.1985, Eliasz 1993]. It also appears that using the

glove side arm to keep the front shoulder closed by adding a linear/angular refinement to

the conventional pitching motion can reduce stress on the anterior throwing shoulder.

These statements need further practical verification in the training process.

Modern technological application of biomechanical principles can be an extremely useful

problem-solving tool for sports medicine professionals. The clinician must be prepared to

make judgments based on objective data when addressing the issue of returning the

injured athlete to his or her sport. Often adjustments can be made in athletic technique to

de-stress the injured body part, either on a temporary or permanent basis. In the case

presented here, permanent change was necessary to allow this athlete to fully return to his

sport and pursue his dream.

DiscussionWith The Subject: Dissecting Problematic Unconventional Marshall

Motion Teaching Cues

Because of the controversial nature of the pitching motion Dr. Marshall has devised it

was interesting to compare specific teaching cues he utilizes with the objectively

measured results they produced in the subject’s technique.

Marshall Instructional Cue: Point the glove arm straight at home plate.

The shoulder line and torso, as found using the JZZ system, is open 30-45º to the

driveline beginning at toe touch, minimizing the total available pelvis and torso rotation.

As demonstrated this leads to loss of hand speed and contributes substantially to release

inconsistency. With this technique premature release of the leading side opens the

shoulders very early and initiates complete rotational movement of the shoulders and

takes away from angular rotation of the shoulders within the linear plane to the target.

Marshall Instructional Cue:Walk forward off the mound.

At toe touch the pelvic line, as taught by Dr. Marshall and as demonstrated by the

subject, is open at stride 30-45º to the driveline, robbing power and minimizing total

rotation before release. To compensate for this Dr. Marshall teaches a 'drop stride' for

specific pitches. It was observed that with this technique the subject’s center of mass

deflects laterally sideways. Biomechanically this was shown to be very inefficient and

very inconsistent. It was observed that the drop step regains degrees of pelvic rotation,

yet at release the torso still has not made up the difference and the arm lags behind

leading to a disconnected linear-rotational-linear kinematic sequence.

Marshall Instructional Cue: Rotate the hips; drive them through and then push off

the stride foot.

It was observed and measured that this cue disrupts the subject’s timing and disconnects

kinematic sequencing. This kinetic disconnect, where the body rotates too fast, too soon,

measurably diminishes hand speeds and resultant ball velocities. The throwing arm and

shoulder is also noted to lag far behind torso rotation and demonstrates great likelihood of

causing a serious anterior shoulder problem. For an athlete with long levers like the

study subject this is particularly problematic.

Marshall Instructional Cue: Punch the throwing hand at the target.

It was observed that this technique disconnects the kinetic chain, resulting in hand-first

movement instead of arm-first movement. It was also observed and documented that

when the hand leads this severely limits hand speed and resultant velocity.

Marshall Instructional Cue: Force couple the glove and throwing hands and lean

back at release.

It was observed that the subject’s lead shoulder opens very early in the kinematic

sequence, resulting in part from premature backward movement of the glove arm. This

action of the glove arm and glove-side torso is observed to contribute significant stress to

the anterior throwing arm shoulder. It was also observed that this simultaneously

exaggerates torso tilt, displacing the subject’s center of gravity laterally, and also

contributes to pre-mature braking of forward movement. The torsional bending observed

in this technique significantly displaces the head as a counterbalance and places

significant stress on the neck and lower back.

Marshall Instructional Cue: Driveline for the ball just above the ear.

This teaching cue was demonstrated to be inaccurate in part because of the significant

upward force applied to the ball hand in the Z-axis. For the forearm to approach vertical

at ball release (highly desirable for spinning the ball in innovative ways), the minimum

driveline is well above the head. If the body rotates far enough, as Dr. Marshall describes,

with the forearm driving vertical (the upper arm then must to be close to horizontal with

very minimal torso tilt) the resultant force is simply a driving forearm arc. This technique

and driveline is not linear as described by Dr. Marshall.

Appendix A: Resources

E-Factor System [JZZ Technologies, Inc. Phoenix, Arizona] tool used to analyze the

above study.

Adams, J.E. (1964). Injury to the throwing arm: A study of traumatic changes in the

elbow joints of the boy baseball players. *California Medicine*, 102(2), 127-132.

Andrews, J. R., Fleisig, G. S., & Whiteside, J. A. (2000) Little leaguer's elbow:

Evaluation, treatment, and prevention. *Sports Medicine Update*, 14(3), 11-15.

Andrews, J. R. & Fleisig, G. S. 1996, How many pitchers should I allow my child to

throw? *USA Baseball News*.

Altchek, D.W. & Dines, D.M. (1995). Shoulder injuries in the throwing athletes. *Journal*, 3(3), 159-165.

of the American Academy of Orthopedic Surgeons

Axe, M. J. (2001). Recommendations for protecting youth baseball pitchers. *Sports*, 9, 147-153.

Medicine and Arthroscopy Review

Axe, M. J., Wickham, R., & Snyder-Mackler, L. (2001). Data-based interval throwing

programs for Little League, high school, college, and professional baseball players.

Sports Medicine and Arthroscopy Review

Axe, M. J., Snyder-Mackler, L., Konin, J. G., & Strube, M. J. (1996). Development of a

distance-based interval throwing program for little league-aged athletes. *The American*, 24(5), 594-602.

Journal of Sports Medicine

Barnett, L. S. (1985). Little league shoulder syndrome: Proximal humeral epiphyseolysis

in adolescent baseball pitchers. *The Journal of Bone and Joint Surgery*, 7-A (3), 495-496.

Carson, W. G., & Gasser, S. (1998). Little Leaguer's shoulder: A report of 23 cases. *The*, 26(4), 575-580.

American Journal of Sports Medicine

Conway, J.E., Jobe, F.W., Glousman, R.E., & Pink, M. (1992). Medial instability of the

Elbow in throwing athletes. *The Journal of Bone and Joint Surgery*, 74-A(1), 67-83.

Dillman, C.J., Fleisig, G.S., & Andrews, J.R. (1993). Biomechanics of pitching with

emphasis upon shoulder kinematics. *Journal of Sport Physical Therapy*, 18(2), 402-408.

Fleisig, G. & Andrews, J. (2002, January 4-5). Effect of pitch type, pitch count and

pitching mechanics on risk of arm pain and injury. Oral presentation at the annual USA

Baseball Medical and Safety Committee Meeting.

Fleisig, G. S., Barrentine, S.W., Escamilla, R.F., & Andrews, J.R. (1996). Biomechanics

of overhand throwing with implications for injuries. *Sports Medicine*, 21(6),421-437.

Gugenheim, J. J., Stanley, R. F., Woods, G. W., & Tullos, H. S. (1976). Little League

Study: the Houston study. *The American Journal of Sports Medicine*, 4(5), 189-200.

Larson, R. L., Singer, K. M., Bergstrom, R., & Thomas, S. (1976). Little league survey:

the Eugene study. *The American Journal of Sports Medicine*, 4(5), 201-209.

Lyman, S. L., Fleisig, G. S., Waterbor, J. W., Funkhouser, E. M., Pulley, L., Andrews, J.

R., et al. (2001). Longitudinal study of elbow and shoulder pain in youth Baseball

pitchers. *Medicine and Science in Sports and Exercise*, 1803-1810.

Lyman, S. L., Fleisig, G. S., Andrews, J. R., & Osinski, E. D. (1998). Youth pitching

injuries: First-ever examination sheds light on arm injuries in youth baseball. *Sports*, 13(2), 4-9.

Medicine Update

Meister, K. (2000). Injuries to the shoulder in the throwing athlete: Part One:

Biomechanics/Pathophysiology/Classification of Injury. *The American Journal Of Sports*, 28(2), 265-275.

Medicine

Pappas, A. M. (1982). Elbow problems associated with baseball during childhood and

adolescence. *Clinical Orthopedics and Related Research*, 164, 30-41.

Rizio, L. & Uribe, J.W. (2001). Overuse injuries of the upper extremity in baseball.

Clinics in Sports Medicine

Werner, S. L., Fleisig, G.S., Dillman, C.J., & Andrews, J.R. (1993). Biomechanics of the

elbow during baseball pitching. *Journal of Sport Physical Therapy*, 17(6), 274-278.

Whiteside, J. A., Andrews, J. R., & Fleisig, G. S. (1999). Elbow injuries in young

baseball players. *The Physician and Sports Medicine*, 27(6), 87-102.

Wilk, K.E, Meister, K., Fleisig, G., & Andrews, J.R. (2000). Biomechanics of the

overhead throwing motion. *Sports Medicine and Arthroscopy Review*, 8, 124-134.

Amin W.K.M., Horyd T., Bober T. (1985): Strength characteristics of team handball

players. In: Biomechanics in Sports II, Terauds J., Barham J.N. /Ed. /. Acad. Publishers,

Del Mar, CA: 379-384.

Atwater A.B. (1980): Biomechanics of overarm throwing movements and of throwing

injuries. Exer. Sport Sci. Rev. 7:43-85.

Bartlett L.R., Storey M.D., Simons B.D. (1989): Measurement of upper extremity torque

production and its relationship to throwing speed in the competitive athlete. Am.J.Sports

Med. 17:89-91.

Best R.J., Bartlett R.M., Morriss C.J. (1993): A three-dimensional analysis of javelin

throwing technique. J. Sports Sci.11:315-328.

Coleman S.G.S., Benham A.S., Northcott S.R. (1993): A three-dimensional

cinematographical analysis of the volleyball spike. J. Sports Sci. 11:295-302.

Dowling J.J., Vamos L. (1993): Identification of kinetic and temporal factors related to

vertical jump performance. J. Appl. Biom. 9:95-110. Eliasz J. (1993): Trening siy

mioeniowej w pice rcznej. Sport Wyczynowy 9/10:21-28.

Eliasz J., Janiak J., Wit A. (1990): Prdkooe lotu piki podczas rzutw w pice rcznej. Sport

Wyczynowy 9/10:17-23.

Feltner M., Dapena J. (1986): Dynamics of the shoulder and elbow joints of the throwing

arm during a baseball pitch. Int. J. Sport Biom. 2:235-259. Filliard J.R. (1985):

Contribution a la mesure de la vitesse du tir en hand-ball. Universite de Paris-Sud, Paris:

2-37.

Jaszczuk J., Buczek M., Karpiowski B., Nosarzewski Z., Wit A., Witkowski M. (1987):

Set-up for measuring in static conditions. Biol. Sport 4:41-55.

Jris H.J.J., Muijen Van E., Ingen Schenau Van G.J., Kemper H.C.G. (1985): Force,

velocity and energy flow during the overarm throw in female handball players. J. Biom.

18:409-414.

Marczinka Z. (1993): Playing Handball. Trio Budapest Publishing Company. I.H.F.

Mikkelsen F., Olesen M.W. (1976): Handbold. Trygg-Hansa, Stockholm. Muijen Van

A.E., Jris H., Kemper H.C.G., Ingen Schenau Van G.J. (1991): Throwing practice with

different ball weights: effects on throwing velocity and muscle strength in female

handball players. Sports Training, Med. Rehab. 2:103-113.

Pauwels J. (1978): The relationship between somatic development and motor ability, and

the throwing velocity in handball for secondary school students. W: Shepard R.J. and

Lavalle H./Ed./ Physical fitness assessment:principles, practice and application,

Springfield, Ill.,Thomas :211-221

Pawlowski D., Perrin D.H. (1989): Relationship between shoulder and elbow isokinetic

peak torque, torque acceleration energy, average power and total work and throwing

velocity in intercollegiate pitchers. Athletic Training 24:129-132.

Pedegana L.R., Elsner R.C., Roberts D., Lang J., Farewell V. (1982): Relationship of

upper extremity strength to throwing speed. Am. J. Sports Med. 10:352-354.

Renne de C., Ho K., Blitzblau A. (1990): Effects of weighted implement training on

throwing velocity. J. Appl. Sport Sci. Res. 4:16-19.

Sakuraj S., Ikegami Y., Okamoto A., Yabe K., Toyoshima S. (1993): A threedimensional

cinematographic analysis of upper limb movement during fastball and

curveball baseball pitches. J. Appl. Biom. 9:47-65.

Whiting W.C., Puffer J.C., Finerman G.A., Gregor R.J., Maletis G.B. (1985): Threedimensional

cinematographic analysis of water polo throwing in elite performers.

Am.J.Sports Med. 13:95-98.

Wooden M., Greenfield B., Johanson M., Litzelman L., Mundrane M., Donatelli R.A.

(1992): Effects of strength training on throwing velocity and shoulder muscle

performance in teenage baseball players. J. Orth. Sports Phys. Therapy 15:223-228. '

1 History

2 Measuring angles

2.1 Units

2.2 Positive and negative angles

2.3 Approximations

3 Types of angle

4 A formal definition

4.1 Using trigonometric functions

4.2 Using rotations

5 Angles between curves

6 The dot product and generalization

7 Angles in Riemannian geometry

9 References

9.1 Notes

9.2 On-line resources

Bones of the Shoulder Girdle/Human anatomy

The Elbow and Radioulnar Joints/Human Anatomy

The Muscle Physiology of Electromyography/Rochville University 2002

Appendix B: Mathematics Utilized In This study

In mathematics, non-Euclidean geometry describes hyperbolic and elliptic geometry,

which are contrasted with Euclidean geometry. The essential difference between

Euclidean and non-Euclidean geometry is the nature of parallel lines. Euclid's 5th

postulate is equivalent to stating that, within a two-dimensional plane, for any given line *l*

and a point *A*, which is not on *l*, there is exactly one line through *A * that does not intersect

l

intersecting *l*, while in elliptic geometry, any pair of lines intersect. (See the entries on

hyperbolic geometry and elliptic geometry for more information.)

Another way to describe the differences between these geometries is as follows: Consider

two straight lines indefinitely extended in a two-dimensional plane that are both

perpendicular to a third line. In Euclidean geometry the lines remain at a constant

distance from each other, and are known as parallels. In hyperbolic geometry they "curve

away" from each other, increasing in distance as one moves further from the points of

intersection with the common perpendicular; these lines are often called ultraparallels. In

elliptic geometry the lines "curve toward" each other and eventually intersect.

A Pair of angles is complementary if the sum of their angles is 90°.

If the two complementary angles are adjacent (i.e. have a common vertex and share a

side, but do not have any interior points in common) their non-shared sides form a right

angle.

In Euclidean geometry, the two acute angles in a right triangle are complementary,

because there are 180° in a triangle and 90° have been accounted for by the right angle.

These angles would be adjacent. On 120 degrees.

Euclid defines a plane angle as the inclination to each other, in a plane, of two lines

which meet each other, and do not lie straight with respect to each other. According to

Proclus an angle must be either a quality or a quantity, or a relationship. The first concept

was used by Eudemus, who regarded an angle as a deviation from a straight line; the

second by Carpus of Antioch, who regarded it as the interval or space between the

intersecting lines; Euclid adopted the third concept, although his definitions of right,

acute, and obtuse angles are certainly quantitative.

Measuring angles

The angle *θ * is the quotient of *s * and *r*.

In order to measure an angle *θ*, a circular arc centered at the vertex of the angle is drawn,

e.g. with a pair of compasses. The length of the arc *s * is then divided by the radius of the

circle *r*, and possibly multiplied by a scaling constant *k * (which depends on the units of

measurement that are chosen):

The value of *θ * thus defined is independent of the size of the circle: if the length of

the radius is changed then the arc length changes in the same proportion, so the ratio

s

In many geometrical situations, angles that differ by an exact multiple of a full circle

are effectively equivalent (it makes no difference how many times a line is rotated

through a full circle because it always ends up in the same place). However, this is

not always the case. For example, when tracing a curve such as a spiral using polar

coordinates, an extra full turn gives rise to a quite different point on the curve.

Units

Angles are considered dimensionless, since they are defined as the ratio of lengths.

There are, however, several units used to measure angles, depending on the choice

of the constant *k * in the formula above.

With the notable exception of the radian, most units of angular measurement are

defined such that one full circle (i.e. one revolution) is equal to *n * units, for some

whole number *n * (for example, in the case of degrees, *n * = 360). This is equivalent to

setting *k * = *n* /2 *π * in the formula above. (To see why, note that one full circle

corresponds to an arc equal in length to the circle's circumference, which is 2 *πr*, so *s*

= 2 *πr*. Substituting, we get *θ * = *ks* / *r * = 2 *πk*. But if one complete circle is to have a

numerical angular value of *n*, then we need *θ * = *n*. This is achieved by setting *k * =

n

The **degree**, denoted by a small superscript circle (°) is 1/360 of a full circle, so

one full circle is 360°. One advantage of this old sexagesimal subunit is that

many angles common in simple geometry are measured as a whole number of

degrees. (The problem of having *all * "interesting" angles measured as whole

numbers is of course insolvable.) Fractions of a degree may be written in normal

decimal notation (e.g. 3.5° for three and a half degrees), but the following

sexagesimal subunits of the "degree-minute-second" system are also in use,

especially for geographical coordinates and in astronomy and ballistics:

The **minute of arc ** (or **MOA**, **arcminute**, or just **minute**) is 1/60 of a

degree. It is denoted by a single prime ( ' ). For example, 3° 30' is equal to

3 + 30/60 degrees, or 3.5 degrees. A mixed format with decimal fractions is

also sometimes used, e.g. 3° 5.72' = 3 + 5.72/60 degrees. A nautical mile was

historically defined as a minute of arc along a great circle of the Earth.

The **second of arc ** (or **arcsecond**, or just **second**) is 1/60 of a minute of

arc and 1/3600 of a degree. It is denoted by a double prime ( ″ ). For

example, 3° 7' 30″ is equal to 3 + 7/60 + 30/3600 degrees, or 3.125 degrees.

θ

The **radian ** is the angle subtended by an arc of a circle that has the same length as

the circle's radius ( *k * = 1 in the formula given earlier). One full circle is 2 *π*

radians, and one radian is 180/ *π * degrees, or about 57.2958 degrees. The radian is

abbreviated *rad*, though this symbol is often omitted in mathematical texts,

where radians are assumed unless specified otherwise. The radian is used in

virtually all mathematical work beyond simple practical geometry, due, for

example, to the pleasing and "natural" properties that the trigonometric functions

display when their arguments are in radians. The radian is the (derived) unit of

angular measurement in the SI system.

The **mil ** is *approximately * equal to a milliradian. There are several definitions.

The **full circle ** (or **revolution**, **rotation**, **full turn ** or **cycle**) is one complete

revolution. The revolution and rotation are abbreviated *rev * and *rot*, respectively,

but just *r * in *rpm * (revolutions per minute). 1 full circle = 360° = 2 *π * rad = 400 gon

= 4 right angles.

The **right angle ** is 1/4 of a full circle. It is the unit used in Euclid's Elements. 1 right

angle = 90° = *π* /2 rad = 100 gon.

The **angle of the equilateral triangle ** is 1/6 of a full circle. It was the unit used by

the Babylonians, and is especially easy to construct with ruler and compasses. The

degree, minute of arc and second of arc are sexagesimal subunits of the Babylonian

unit. 1 Babylonian unit = 60° = *π* /3 rad ≈ 1.047197551 rad.

The **grad**, also called **grade**, **gradian**, or **gon ** is 1/400 of a full circle, so one full

circle is 400 grads and a right angle is 100 grads. It is a decimal subunit of the right

angle. A kilometer was historically defined as a centi-gon of arc along a great circle

of the Earth, so the kilometer is the decimal analog to the sexagesimal nautical mile.

The gon is used mostly in triangulation.

The **point**, used in navigation, is 1/32 of a full circle. It is a binary subunit of the full

circle. Naming all 32 points on a compass rose is called "boxing the compass".

1 point = 1/8 of a right angle = 11.25° = 12.5 gon.

The astronomical **hour angle ** is 1/24 of a full circle. The sexagesimal subunits were

called **minute of time ** and **second of time ** (even though they are units of angle).

1 hour = 15° = *π* /12 rad = 1/6 right angle ≈ 16.667 gon.

The **binary degree**, also known as the **binary radian ** (or **brad**), is 1/256 of a full

circle. The binary degree is used in computing so that an angle can be efficiently

represented in a single byte.

The **grade of a slope**, or **gradient**, is not truly an angle measure (unless it is

explicitly given in degrees, as is occasionally the case). Instead it is equal to the

tangent of the angle, or sometimes the sine. Gradients are often expressed as a

percentage, for the usual small values encountered (less than 5%), the grade of a

slope is approximately the measure of an angle in radians.

Positive and negative angles

A convention universally adopted in mathematical writing is that angles given a sign

are **positive angles ** if measured counterclockwise, and **negative angles ** if measured

clockwise, from a given line. If no line is specified, it can be assumed to be the xaxis

in the Cartesian plane. In many geometrical situations a negative angle of − *θ * is

effectively equivalent to a positive angle of "one full rotation less *θ* ". For example, a

clockwise rotation of 45° (that is, an angle of −45°) is often effectively equivalent to

a counterclockwise rotation of 360° − 45° (that is, an angle of 315°).

In three dimensional geometry, "clockwise" and "counterclockwise" have no

absolute meaning, so the direction of positive and negative angles must be defined

relative to some reference, which is typically a vector passing through the angle's

vertex and perpendicular to the plane in which the rays of the angle lie.

In navigation, bearings are measured from north, increasing clockwise, so a bearing

of 45 degrees is north-east. Negative bearings are not used in navigation, so northwest

is 315 degrees.

Approximations

1° is approximately the width of a pinky finger at arm's length

10° is approximately the width of a closed fist at arm's length.

20° is approximately the width of a hand span at arm's length.

Types of angle

An angle of 90° ( *π* /2 radians, or one-quarter of the full circle) is called a **right angle**.

Two lines that form a right angle are said to be **perpendicular ** or **orthogonal**.

Angles smaller than a right angle (less than 90°) are called **acute angles ** ("acute"

meaning "sharp").

Angles larger than a right angle and smaller than two right angles (between 90° and

180°) are called **obtuse angles ** ("obtuse" meaning "blunt").

Angles equal to two right angles (180°) are called **straight angles**.

Angles larger than two right angles but less than full circles (between 180° and

360°) are called **reflex angles**.

Angles that have the same measure are said to be **congruent**.

Two angles opposite each other, formed by two intersecting straight lines that form

an "X" like shape, are called **vertical angles ** or **opposite angles**. These angles are

congruent.

Angles that share a common vertex and edge but do not share any interior points are

called **adjacent angles**.

Two angles that sum to one right angle (90°) are called **complementary angles**.

The difference between an angle and a right angle is termed the **complement ** of

the angle.

Two angles that sum to a straight angle (180°) are called **supplementary angles**.

The difference between an angle and a straight angle is termed the **supplement ** of

the angle.

Two angles that sum to one full circle (360°) are called **explementary angles ** or

conjugate angles

The smaller angle at a point where two line segments join is called the **interior**.

angle

In Euclidean geometry, the measures of the interior angles of a triangle add up to

π

add up to 2 *π * radians, or 360°. In general, the measures of the interior angles of a

simple polygon with *n * sides add up to [( *n * − 2) × *π* ] radians, or [( *n * − 2) × 180]°.

The angle supplementary to the interior angle is called the **exterior ** angle.

The angle between two planes (such as two adjacent faces of a polyhedron) is called

a **dihedral angle**. It may be defined as the acute angle between two lines normal to

the planes.

The angle between a plane and an intersecting straight line is equal to ninety degrees

minus the angle between the intersecting line and the line that goes through the point

of intersection and is normal to the plane.

If a straight transversal line intersects two parallel lines, corresponding (alternate)

angles at the two points of intersection are congruent; adjacent angles are

supplementary (that is, their measures add to *π * radians, or 180°).

Formal definition Using trigonometric functions

A Euclidean angle is completely determined by the corresponding right triangle. In

particular, if θ is a Euclidean angle, it is true that

and

for two numbers *x * and *y*. So an angle in the Euclidean plane can be

legitimately given by two numbers *x * and *y*.

To the ratio there correspond two angles in

the geometric range 0 < θ < 2π, since '

Suppose we have two unit vectors and

in the euclidean plane

. . Then there exists one positive isometry (a

rotation), and one only, from to

that maps *u * onto *v*. Let *r * be such a rotation. Then

the relation defined by

is an equivalence relation and we call

angle of the rotation *r * the equivalence class

, where denotes the

unit circle of . . The angle between two vectors

will simply be the angle of the rotation that maps one onto the other.

We have no numerical way of determining an angle yet. To do this,

we choose the vector (1,0), then for any point M on

at distance θ from (1,0) (on the circle), let

. . If we call *r* θ the rotation that

transforms (1,0) into , , then

is a bijection, which means we can

identify any angle with a number between 0 and 2π.

Angles between curves

The angle between the two curves is defined as the angle between the

tangents *A * and *B * at *P*

The angle between a line and a curve (mixed angle) or between two

intersecting curves (curvilinear angle) is defined to be the angle

between the tangents at the point of intersection. Various names (now

rarely, if ever, used) have been given to particular cases:—

amphicyrtic

* (Gr. *ἀμφί*, on both sides, *κυρτόσ*, convex) or *cissoidal

(Gr. *κισσόσ*, ivy), biconvex; *xystroidal * or *sistroidal * (Gr. *ξυστρίσ*, a

tool for scraping), concavo-convex; *amphicoelic * (Gr. *κοίλη*, a

hollow) or *angulus lunularis*, biconcave.

The dot product and generalization

In the Euclidean plane, the angle θ between two vectors **u ** and **v ** is

related to their dot product and their lengths by the formula

This allows one to define angles in any real inner product space,

replacing the Euclidean dot product · by the Hilbert space inner

product <·,·>.

Angles in Riemannian geometry

In Riemannian geometry, the metric tensor is used to define the

angle between two tangents. Where *U * and *V * are tangent vectors

and *gij * are the components of the metric tensor *G*,

Using rotations

* radians, or 180°; the measures of the interior angles of a simple quadrilateral'*.

* = *s* / *r * rad = 1 rad.* /2 *π*.)

* / *r * is unaltered.*. In hyperbolic geometry, by contrast, there are infinitely many lines through *A * not

Contents

*, 20(3), 453-468.*, 9, 24-34.

Conclusions & Outcomes

Refined Conventional Motion: Body Posture at Toe Touch

Refined Conventional Motion: Rotational Speeds and Progressive Speed Gains

Refined Conventional Motion: Analysis Summary