Reflection In an Arbitrary Surface

[Jeff.N]

Reflection in a spherical mirror behaves like inversion in the sphere, which is the 3D equivalent to inversion in a circle.

2D Reflections in circles [i think, just play along] can be generalized to arbitrary 2D curves by finding the normal line(s) through the curve passing through the point we wish to reflect and inverting that point in the respective circle(s) of curvature.

Can This method be generalized to reflection in surfaces in 3D?

 

[chrisbaird]

Reflections of light rays off of arbitrarily-shaped mirror surfaces is found by ray-tracing: following a ray until it intersects the object, applying the law of reflection at that point, following the ray again until it intersects the object again, etc.

 

[Jeff.N]

How does one find the image of a point under reflection in an arbitrary surface.

the image of a point X reflected in a plane P is the point X' on the normal line to P through X such that distance(X,P)=distance(X',P) and X(!=)X'

the image of a point X reflected in a sphere S, X', is the inversion of the point X in the sphere S, (assuming the sphere is of radius R and is centered at O). X' is the point on the ray OX such that |O-X||O-X'|=R^2

reflection in a line is the limiting case of reflection in the sphere, as the radius of the sphere tends to infinity and the distance from the point we wish to reflect to the center of sphere tends to infinity.

 

[A.T.]

How does one find the image of a point under reflection in an arbitrary surface.

For an arbitrary surface there is no general solution. A single point could have multiple images.

 

[Jeff.N]

For an arbitrary surface there is no general solution. A single point could have multiple images.

The same is true in 2D for an arbitrary curve, however, that didn't stop me.

"2D Reflections in circles be generalized to arbitrary 2D curves by finding the normal line(s) through the curve passing through the point we wish to reflect and inverting that point in the respective circle(s) of curvature."

Is there a reason this can be done in 2d and not 3d?

my main problem when trying to use a similar argument in 3d is that different curves on the same surface through the same point can have different curvatures at that point.

edit
i think that if you could reflect in the quadratic form a surface approximates at a point given by the second fundamental form at each point.

so now my question is how would one determine the image of a point in a mirror on the surface of a quadratic form.