Unknown Part B Geometry

Question 1

Ambient Distance

Answer 1

Euclidean Distance between points A and B

Question 2

Intrinsic Distance

Answer 2

Length of the shorter arc if a great circle joining A and B

Question 3

What is an isometry and how is it affected across ambient and intrinsic distance

Answer 3

an isometry preserves distance and does not differ between intrinsic and ambient distances (gives the same distances)

Question 4

When do ambient isometries restrict to sphere

Answer 4

when the isometry sends to itself, preserves distance from centre O and therefore must fix to O

Question 5

In Ambient 3 space, rotations restrict to isometries when

Answer 5

rotations about a line preserve distance from that line
restrict to isometry of sphere if axis pass through centre
- sphere preserves exactly in this case

Question 6

reflection restrict to isometry when

Answer 6

reflection preserves distance from plane
restrict if plane passes through the centre
otherwise points will move off the sphere

Question 7

Translation restrict to isometries when

Answer 7

moves centre off sphere

sphere not preserved unless translation is 0
only the identity translation works.