Page 68 - Standard Handbook Of Petroleum & Natural Gas Engineering

Analytic Geometry 57

Equations of a Line

Intersection of two planes
alx + bly + cIz + d, = 0
i apx + b,y + c,z + x, = 0
For this line

Symmetric form, Le., through (x,,yI,zI) with direction numbers a, b, and c
(x - xJ/a = (y - YJ/b = (z - z,)/c
Through two points

x--1 - Y-Y1 - Z-Z1
X2-Xx1 YZ-YI 22-21
where h:p:v = (xp - xI):(y2 - y1):(z2 - zl)
Equations of Angles
Between two lines
cos e = h,h, + plp2 + vIv2
and the lines are parallel if cos 8 = 1 or perpendicular if cos 8 = 0
Between two planes, given by the angle between the normals to the planes.

Equation (standard form) of a Sphere (Figure 1-44)
x‘L + y2 + 22 = r

Equation (standard form) of an Ellipsoid (Figure 1-45)

V- V
Figure 1-44. Sphere. Figure 1-45. Equation of an ellipsoid.