Post by Herbert Blenner on Jan 15, 2019 13:03:22 GMT -5
by Herbert Blenner | Posted November 7, 2007
A circular or elliptical hole shows entry by a bullet with a negligible yaw angle. Under these conditions the circular displacement area of the bullet punches a cylindrical wound track beneath the surface. When the direction of a bullet with proper alignment is perpendicular to the struck surface, the hole is circular. More commonly, the trajectory of the bullet makes an angle, known as incidence, with the perpendicular to the surface. Although the bullet still punches a cylindrical wound track, the surface hole is a special type of oval known as an ellipse. Two mutually perpendicular axes characterize an ellipse. These axes, known as the minor and the major, divide the ellipse into two symmetrical portions. For a rigid material, plane geometry proves that the length of the minor axis divided by the length of the major axis equals the cosine of the incidence angle. This relationship enables a forensic analyst to place the trajectory of the bullet on the surface of a cone whose axis is perpendicular to the struck surface and apex coincides with the point of entry. Analytic geometry proves that the major axis lies in the same plane as the striking velocity of the bullet and the length of the minor axis equals the diameter of the bullet. The former result completes the description of the striking angles of the bullet and limits number of trajectories to two. The latter result enables an analyst who knows the diameter of the bullet to adapt the simple analysis of rigid materials to deformable materials, such as tissue or bone.
Figure 1 - Bullet holes of entrance
James J. Humes did not quantify the orientation angles of the major axes of the bullet holes. Instead he verbally described the major axis of the hole in the back as roughly parallel to the vertical column. So Commission Exhibit 386, CE 386, has increased importance. It shows that the major axis of the bullet hole in the head made a 18-degree clockwise angle with the direction of the vertical column. For the hole in the back, the major axis made an angle of -15 degree with the vertical column. The purpose of using positive angles for clockwise rotations is to be consistent with the unconventional practice of the Forensic Pathology Panel to reference directions to an imaginary clock. The angular orientation of the major axis gives the azimuthal component of the striking angles of the bullet. This information confines the bullet to two possible trajectories. One trajectory has a positive incidence angle and the other has a negative incidence angle.
When prosectors do not dissect a wound track, forensic analysts choose from several techniques to resolve ambiguity of the algebraic sign. If the abrasion is oval, then they select the trajectory, which makes an acute angle with the more prominent portion of the abrasion. When the incidence angle is sufficiently large to show undermining, then they choose the trajectory that makes an obtuse angle to the undermined portion of the wound. A third method selects the trajectory in the direction of the shallower to the deeper portion of the surface hole. Regrettably, the prosectors did not provide the details to resolve the ambiguous sign.
James J. Humes described the bullet hole in President Kennedy's back as oval with a major axis of 7 mm and a minor axis of 4 mm. He explained that a tangential strike by the bullet elongated the wound and attributed the length of the minor axis being less than the diameter of the 6.5 mm bullet to elastic recoil of the skin. This latter observation shows that James J. Humes gave the dimensions of the bullet hole, which he called a wound. This information is sufficient to calculate the incidence angle of the bullet if oval accurately described the shape.
Distance along the wound track accounts for the elongation of the major axis. Specifically the square of this distance equals ( 7 mm ) 2 - ( 4 mm ) 2 or 33 mm 2. When elastic relaxation and swell of tissues have negligible effect upon this length, the square of the unreduced length of the major axis, b, minus the square of the unreduced length of the minor axis, a, equals 33 mm 2, where a is the 6.5 mm diameter of the bullet. Hence b 2 - ( 6.5 mm ) 2 = 33 mm 2. Solving this equation yields b = 8.7 mm. The cosine of the incidence angle equals the length of the unreduced minor axis divided by the length of the unreduced major axis. Thus, the angle of incidence is ±42 degree.
The reported dimensions of the bullet hole are rounded to one significant figure. So the actual length of the major axis should be taken as 7 ±0.5 mm. Likewise the real length of the minor axis is 4 ±0.5 mm. These uncertain dimensions produce a span for the magnitude of incidence angle. Calculations yield 41 ±5 degree.
When the trajectory has negligible curvature, the angle between the geographic horizontal and the initial direction of the wound track is the declination angle of the bullet. By definition incidence is the angle between the direction of the perpendicular to the entry site and the initial direction of the wound track. So the angle between the direction of the horizontal and the perpendicular equals the incidence angle minus the declination angle. This difference of the two angles also equals the angle between the geographic vertical and the direction of a tangent to the surface at the entry site. For a declination angle of 20 degree and an incidence angle of +42 degree, the tangent to the surface makes an angle of -22 degree with the vertical. This case represents a 22-degree recline angle. Alternately the negative incidence angle, yields a difference of 20 degree minus -42 degree or 62 degree as the angle between the vertical and a leaning surface. These results are amenable to geometric proof.
An analytic solution of this problem in three dimensions changes the angles of recline and lean by less than two degrees and justifies the two-dimensional simplification of this particular situation.
Figure 2 - Testing lower entry sites
A modification of CE 385 shows that lowering the entry site of the back wound decreases the incidence angle. Point H is the original entry site on CE 385 and AH represents the original trajectory with a 12-degree clockwise rotation of the entire graphic to give the trajectory an apparent 20-degree declination angle. This rotation explicitly shows the Warren Commission explanation of the bullet holes in the back and neck of President Kennedy.
Constructing the perpendicular, EH, at point H enables measuring the angle of incidence, AHE, as 27 degree. Under these conditions a 6.5-mm bullet with negligible yaw angle punches an elliptical surface hole with an unreduced minor axis of 6.5 mm and an unreduced major axis of 6.5 mm / cos 27 degree or 7.3 mm. Distance along the wound track accounts for elongation of the major axis. The square of this unreduced distance equals (7.3 mm ) 2 - ( 6.5 mm ) 2 or 11 mm 2. Since the effects of elastic relaxation and swell of tissue upon the contributory distance along the wound track are small, the unreduced distance is approximately equal to the reduced distance. So the reduced length of the major axis, b', and the reduced length of the minor axis, a', satisfy the relationship, b' 2 - a' 2 ~ 11 mm 2. Taking a' = 4 mm gives b' ~ 5.2 mm. This elongation of the major axis by 30% is a mere 40% of the 75% elongation reported by James J. Humes.
Repeating the above procedure at lower points of entry J, K and L show that the incidence angle decreases at point L, becomes zero at point K and increases at the lowest test point L. Since distances between these trial points are negligible compared with the height of the shooter above the back, the respective trajectories BJ, CK and DL are essentially parallel.
Curvature of the back changes the directions of the respective perpendiculars, FJ, CK and GL. The resultant angles of incidence, BJF = 16 degree, CKC = 0 degree and DLG = -11 degree yield 4.4 mm, 4.0 mm and 4.2 mm as the approximate reduced lengths of the major axes. These results show the irreconcilable conflict between the autopsy description of a 7 mm by 4 mm oval bullet hole in the back of President Kennedy and the Warren Commission explanation of the back wound.