On the reflection point where light reflects to a known destination on quadratic surfaces

Abstract

We address the problem of determining the reflection point on a specular surface where a light ray that travels from a source to a target is reflected. The specular surfaces considered are those expressed by a quadratic equation. So far, there is no closed form explicit equation for the general solution of this determination of the reflection point, and the usual approach is to use the Snell law or the Fermat principle whose equations are derived in multidimensional nonlinear minimizations. We prove in this Letter that one can impose a set of three restrictions to the reflection point that can impose a set of three restrictions that culminates in a very elegant formalism of searching the reflection point in a unidimensional curve in space. This curve is the intersection of two quadratic equations. Some applications of this framework are also discussed.

A Framework for Augmented Reality using Non-Central Catadioptric Cameras

“…Now, for all the vertices of the triangles (j ) (i) p (C ) (in the coordinates of the camera system), the goal is to compute the reflection point in the mirror (j ) (i) r (C ) . We used the solution method proposed by Gonçalves [22].…”

“…For central cameras, this direction is computed by considering viewer's position at the "single view point". To solve this problem, we define the viewer's position at the respective reflection point on the mirror, which can be computed using [22,23]. Fig.…”

“…Thus, to project these 3D triangles, one just needs to take into account the projection of three 3D points (that form the vertices of the triangles). The forward projection of 3D points for images of non-central catadioptric cameras was addressed by Gonçalves [22] and Agrawal [23].…”

A Framework for Augmented Reality Using Non-Central Catadioptric Cameras

“…Now, for all the vertices of the triangles pjq piq p pCq (in the coordinates of the camera system), the goal is to compute the reflection point in the mirror pjq piq r pCq . We follow the solution "QI Projection" method proposed by Gonçalves [16]. We note that other solutions could be used, for instance the method proposed by Agrawal et al [17].…”

“…As a result, to project these 3D triangles we just need to take into account the projection of three 3D points (that form the vertices of the triangles) to the 2D image plane. This problem was addressed by Gonçalves [16] (which denoted the problem as "QI Projection") and Agrawal [17]. Since the geometry of these imaging systems does not verify most properties of the conventional perspective cameras, we also had to reformulate conventional approaches to other problems, such as occlusions and object illumination.…”