Reflections on spheres and cylinders of revolution
Georg Glaeser1999
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Abstract
In computer graphics, it is often an advantage to calculate reflections directly, especially when the application is time-critical or when line graphics have to be displayed. We specify formulas and parametric equations for the reflection on spheres and cylinders of revolution. The manifold of all reflected rays is the normal congruence of an algebraic surface of order four. Their catacaustic surfaces are given explicitly. The calculation of the reflex of a space point leads to an algebraic equation of order four. The up to four practical solutions are calculated exactly and efficiently. The generation of reflexes of straight lines is optimized. Finally, reflexes of polygons are investigated, especially their possible overlappings. Such reflexes are the key for the reflection of polyhedra and curved surfaces. We describe in detail how to display their contours.
Key takeaways
- In this paper we proceed the other way round ("backward ray tracing", [1], [22]): Given a surface Φ, the eye point E and an arbitrary space point S, we are looking for a light ray s S that runs through E after being reflected on Φ in a "reflex" R ∈ Φ.
- We thus have the following theorem about the possible degeneration of the set of reflected points: Theorem 2: The set R[Φ κ ; E](S) contains the two reflexes on M E. When non-trivial real reflexes occur, the reflex additionally consists of a small circle of Φ κ with the axis M E.
- When reflecting points outside of Φ, the corresponding distance of the light ray is minimal for the reflection on the outside of Φ.
- When b intersects the axis M E the sphere Φ κ , the curve degenerates into two circles: One consists of the reflexes of all points of b except the intersection point.
- Theorem 8: When reflecting on a sphere Φ κ , the reflex of a polygon does not overlap when it's carrier plane does not intersect Φ κ .
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