1. Parabolic mirrors have the nice focusing property that allrays coming in parallel to their axis of symmetry are focused tothe same point. Suppose that we have a parabolic mirror surfacethat is implicitly dened by the equation F(x, y, z) = 0, where
F(x; y; z) = x^2 + y^2 - z :
(a) Use the gradient of F(x, y, z) to find a vector normal tothe surface at the point (1, 2, 5).
(b) What is the standard equation for the plane tangent, T, toour paraboloid at (1, 2, 5)? What isthe normal vector to T?
(c) Consider a point on a surface with a local normal n, and aray of light that reflects of the surface at that point. Let thedirection of the incoming and reflected rays be win andwout,respectively. The formula for nding the direction of thereflected ray, given the incoming rayand the localnormal, \(w_{out}=w_{in}-2((w_{in} dot n) / (n dot n))*n\)
n. Draw a picture and explain why this gives us vectors wherethe angle of incidence equals the angle of reflection.
(d) Consider an incoming ray that is parallel to the z-axis sothat a vector in the direction of this
ray is win = -k. What is wout if win intersects the mirror at(1; 2; 5)?
(e) What is the parameterization for the line, L, thatintersects the surface at the point (1; 2; 5)
and is in the direction of wout?
(f) The point where L intersects the z-axis is the focal pointof the parabolic mirror. Determine the coordinates of the focalpoint for our mirror.
1. Parabolic mirrors have the nice focusing property that allrays coming in parallel to their axis of symmetry are focused tothe same point. Suppose that we have a parabolic mirror surfacethat is implicitly dened by the equation F(x, y, z) = 0, where
F(x; y; z) = x^2 + y^2 - z :
(a) Use the gradient of F(x, y, z) to find a vector normal tothe surface at the point (1, 2, 5).
(b) What is the standard equation for the plane tangent, T, toour paraboloid at (1, 2, 5)? What isthe normal vector to T?
(c) Consider a point on a surface with a local normal n, and aray of light that reflects of the surface at that point. Let thedirection of the incoming and reflected rays be win andwout,respectively. The formula for nding the direction of thereflected ray, given the incoming rayand the localnormal, \(w_{out}=w_{in}-2((w_{in} dot n) / (n dot n))*n\)
n. Draw a picture and explain why this gives us vectors wherethe angle of incidence equals the angle of reflection.
(d) Consider an incoming ray that is parallel to the z-axis sothat a vector in the direction of this
ray is win = -k. What is wout if win intersects the mirror at(1; 2; 5)?
(e) What is the parameterization for the line, L, thatintersects the surface at the point (1; 2; 5)
and is in the direction of wout?
(f) The point where L intersects the z-axis is the focal pointof the parabolic mirror. Determine the coordinates of the focalpoint for our mirror.
