Mathematic Background

The mathematic definition of the paraboloid that we will be working with is:

 for

Look at Figure 1 to see how this equation is evaluated at various values of x and y. This equation is going to be used throughout this section to provide a mathematic basis for the implementation that we will be doing later on.

The first task at hand is to be able to find the normal vector at any point on the paraboloid surface. We will use the relatively well-known method of finding two vectors that are tangential to the surface and taking their cross product to produce the normal vector. The two tangent vectors are found by taking the partial derivatives of the function with respect to x and y. So here is the mathematic version:

With the normal vector now known for the entire paraboloid surface, we can solve for the x and y coordinates of the intersection of an incoming ray and the paraboloid surface. This result will later be used to determine the x and y texture coordinates for both generating and accessing the paraboloid maps.

To find the intersection point on the paraboloid, we must have the direction of the incident ray as well as the direction of the reflected ray. Since the paraboloid reflects all rays to the forward direction for the forward hemisphere, then we know the reflected direction – it is the same for all reflections in that hemisphere!  Examine figure 2 if this is not clear to you, it is crucial to understand. So the forward hemisphere's reflection vector is always going to be (0,0,1) and the backward hemisphere's reflection vector is always going to be (0,0,-1).

Now that we have both the incident vector and the reflected vector, we can calculate the normal vector that caused the reflection. The sum of the incident and reflected vectors should have the same direction as the normal vector, although with a different magnitude (assuming the vectors are the same length!). Again, in math form:

Then from our earlier derivation of the normal vector:

This relationship shows that if we divide the result of the sum of the two vectors by their z component, then the resulting x and y components are the x and y components of the paraboloid surface that reflected the vector.

The relationships that we have defined in this section will allow us to implement the paraboloid mapping in HLSL.

Generating Paraboloid Maps

Now that we have a better understanding of the inner workings of the paraboloid parameterization, it is time to put our newly acquired knowledge to use. First we will look at how to generate a dual paraboloid environment map, and then we will look at how to access the map for use in the various shaders that can benefit from it. The sample effect files are written in HLSL, but the implementation should apply to other shading languages as well.

To begin, the application must calculate the transformation matrices to be used in our rendering pipeline. I will quickly discuss the three basic transforms usually used: the world, view, and projection matrices.

The world matrix remains the same as with standard rendering – it just positions the object in world space and is generated in the normal fashion.

The view matrix is generated in the normal fashion as well, but the 'camera' that will be used to create it from is really the point P. So to create the view matrix, the world space position of P will be used as the translation part, and an appropriate orientation has to be selected to provide the rotation part. The orientation that is used will determine what will end up being the forward and backward directions for the two paraboloids.

The projection matrix is actually not needed to generate the paraboloid map. The input geometry will be transformed into camera space and its position will be directly manipulated in the vertex shader. In this example the projection matrix is simply set to the identity matrix.

Once all three matrices have been calculated, we are ready to start rendering our geometry. The majority of the work in producing a paraboloid map is done in the vertex shader. You can think of the overall algorithm that we are trying to implement as requiring us to place each vertex into a location in the paraboloid map that can later be looked up according to the mathematic rules that we defined earlier.

The first operation needed is to transform the incoming vertices with the combined world, view, and projection matrix object space to post projection space and divide by the w coordinate to produce a homogenous position in camera space.

OUT.position = mul( float4(IN.position, 1), mWVP );
OUT.position = OUT.position / OUT.position.w;

To find the x and y coordinates of the map that we should store this vertex at, we will need to have a vector from the point P to the transformed vertex. In the paraboloid basis (set by the view matrix earlier) point P is actually located at (0,0,0) so finding a vector from P to the vertex is as simple as dividing P by the length of the vector from (0,0,0) to P. 

float L      = length( OUT.position.xyz );
OUT.position = OUT.position / L;

The resulting vector represents a ray from the vertex toward point P. Now we must find the x and y coordinates of the point where this ray intersects the paraboloid surface. From the earlier findings, the normal vector at any point on the paraboloid surface can be found by adding the incident and reflected vectors, and then dividing that result by its z component. We know that the reflected vector is always (0,0,1) due to the properties of the paraboloid (see Figure 2). So to compute the normal vector, we only have to add 1 to the z component and then divide the x and y components by this new z value.

OUT.position.z = OUT.position.z + 1;
OUT.position.x = OUT.position.x / OUT.position.z;
OUT.position.y = OUT.position.y / OUT.position.z;

This x and y coordinates represent the positions in the output texture that this vertex maps to in the paraboloid parameterization. To have proper depth testing, I have set the z coordinate back to the distance from P to the vertex in world space scaled by the viewing distance. In addition, the w component is set to 1 to finalize the x, y, and z values output by the vertex shader.

OUT.position.z = (L - 0.1)/(500.0-0.1);
OUT.position.w = 1;

The only change necessary between rendering the front map and the back map is to multiply the z coordinate by –1 before dividing by the distance to the vertex. Also, back face culling is disabled in the sample file so that the same effect can be used on both maps (by negating z the triangle winding would be reversed). This could also be done with two different techniques, but I thought it would be simpler this way.

To see a complete listing of the vertex shader that we have built over the last few paragraphs as used in generating a two texture terrain, see the provided effect file: dual_paraboloid_terrain.fx

Here are some sample paraboloid maps generated from a Terragen terrain file:

Figure 3: Sample Front and Back Paraboloid Maps

Figure 4: Wireframe Views of Sample Front and Back Paraboloid Maps

Accessing Paraboloid Maps