Question

Let S denote the sphere x2 y2 2 = 1. Given two points P(1,0,0), (a) Find the distance between P and Q. Lets call this Euclidean distance. (b) Find the plane that goes through O, P, Q. What is the intersection of this plane with the sphere? (Hint: use OP × OQ as the the normal vector) (c) Observe that the length of the arc PQ is 0 the angle between OP,0Q in radians. (Hint: You know how to find the circumference of a circle, this arc is a part of a circle) (d) Find the angle θ. Make sure your calculator is in rad option. That gives the length of the arc. This distance is called the spherical dis- tance between P and Q. (If we do not allow leaving the surface of the sphere, length of the shortest path between P and Q is spherical dis- tance (e) Compare euclidean and spherical distance between P and Q. Which one is larger?

Answer 1

Let S denote the sphere x2 + y2-22 = 1. Given two points P(1,0,0), Q(-,-,-) (a) Find the distance between P and Q. Lets call this Euclidean distance. Ans.: P(1,0,0) and Q =は-1)は一0] +は-0) Euclidean distance +1+-= 2 0.919402 (b) Find the plane that goes through O,P,Q. What is the intersection of this plane with the sphere? (Hint: use OP x OQ as the the normal vector) Ans. 0(0,0,0), P(10,0), and OP= (1-0)i+(0-0))+(0-0)k-1

n=OPs0Q =| 1 0|=1(0-0)-j 0 -叶下z=0 4By+で- y-z=0 (c) Observe that the length of the arc PQ is θ the angle between OP,OQ in radians. (Hint: You know how to find the circumference of a circle, this arc is a part of a circle) (d) Find the angle θ. Make sure your calculator is in rad option. That gives the length of the arc. This distance is called the spherical dis- tance between P and Q. (If we do not allow leaving the surface of the sphere, length of the shortest path between P and Q is spherical dis- tance.) (e) Compare euclidean and spherical distance between P and Q. Which one is larger? Ans.:

-1 θ=cos 0.9553 V 3 Length of the arc PC-r9-1 × 0.9553 = 0.9553 → Spherical distance Spherical distance is larger.