Math 445 Assignment 5A (due FRIDAY 1/31)

5-1 Intersecting a certain line with a sphere

Let N = (0,0,1) and let the plane F be the (x,y,0) plane (i.e., the z = 0 plane).

• For any point Q = (a, b, 0) in F, write the points of line NQ in parametric form. (i.e., in terms of an affine parameter t).

(1-t)N + tQ = (ta, tb, 1-t)

• Let E be the sphere with center (0,0,0) and radius 1.  Write the equation in x, y, z that defines this sphere.

x² + y² + z² = 1

• By substituting the points on NQ into the equation for E, solve for t and find the points of intersection of NQ with E.  The answers will be in terms of a and b.

t²(a²+b²) + (1-t)² = 1

t²(a²+b²) + 1 - 2t + t² = 1

t²(1+ a²+b²) - 2t = 0

(t(1+ a²+b²) - 2)t = 0

Roots are t = 0 and t = 2/(1+ a²+b²)

Points of intersection are N = (0,0,1) for t = 0 and

P = (1/(1+ a²+b²))[2a, 2b, a² + b² - 1]