Dual Paraboloid Mapping

Definition

Let the equation of the (elliptic) paraboloid be z = 1 2 − 1 2 ( x 2 + y 2 ) \displaystyle z = \frac{1}{2} - \frac{1}{2}(x^2 + y^2) .

Evidently, the plane section of this paraboloid is the parabola of which the focus is the origin, the vertex is ( 0 , 0 , 1 2 ) \displaystyle (0, 0, \frac{1}{2}) and the directrix is z = 1 \displaystyle z = 1 .
Proof:
For example, let y = 0 \displaystyle y = 0 , we have the intersection of the paraboloid and ZOX plane z = 1 2 − 1 2 x 2 \displaystyle z = \frac{1}{2} - \frac{1}{2} x^2 which can be obtained by translating parabola z = − 1 2 x 2 \displaystyle z = - \frac{1}{2} x^2 by 1 2 \displaystyle \frac{1}{2} units in the positive Z direction.
And the parabola z = − 1 2 x 2 \displaystyle z = - \frac{1}{2} x^2 can be written as x 2 = − 2 z \displaystyle x^2 = -2z . Compared to the standard equation x 2 = − 2 p z \displaystyle x^2 = -2pz , we have semi-latus rectum p = 1 \displaystyle p = 1 . This means that z = − 1 2 x 2 \displaystyle z = - \frac{1}{2} x^2 is the parabola of which the focus is the ( 0 , 0 , − p 2 ) = ( 0 , 0 , − 1 2 ) \displaystyle (0, 0, -\frac{p}{2}) = (0, 0, -\frac{1}{2}) , the vertex is the origin and the directrix is z = p 2 = 1 2 \displaystyle z = \frac{p}{2} = \frac{1}{2} .
And we translate parabola z = − 1 2 x 2 \displaystyle z = - \frac{1}{2} x^2 by 1 2 \displaystyle \frac{1}{2} units in the positive Z direction to obtain z = 1 2 − 1 2 x 2 \displaystyle z = \frac{1}{2} - \frac{1}{2} x^2 of which the focus is the origin ( 0 , 0 , − p 2 + 1 2 ) = ( 0 , 0 , 0 ) \displaystyle (0, 0, -\frac{p}{2} + \frac{1}{2}) = (0, 0, 0) , the vertex is ( 0 , 0 , 0 + 1 2 ) = ( 0 , 0 , 1 2 ) \displaystyle (0, 0, 0 + \frac{1}{2}) = (0, 0, \frac{1}{2}) and the directrix is z = 1 2 + 1 2 = 1 \displaystyle z = \frac{1}{2} + \frac{1}{2} = 1 .

Let ( x_norm , y_norm , z_norm ) → \displaystyle \overrightarrow{(\text{x\_norm}, \text{y\_norm}, \text{z\_norm})} be one of the directions in the domain of the spherical function. The goal of the paraboloid mapping is to transform this direction to the 2D texture coordinate ( x_2d , y_2d ) \displaystyle (\text{x\_2d}, \text{y\_2d}) .
Proof:

Let the ray from focus F with direction ( x_norm , y_norm , z_norm ) → \displaystyle \overrightarrow{(\text{x\_norm}, \text{y\_norm}, \text{z\_norm})} intersect the paraboloid at point P. Evidently, the unit vector in the direction of F P → \displaystyle \overrightarrow{FP} is ( x_norm , y_norm , z_norm ) → \displaystyle \overrightarrow{(\text{x\_norm}, \text{y\_norm}, \text{z\_norm})} .
Let PM be the perpendicular distance from P to directrix z = 1 \displaystyle z = 1 . Evidently, the unit vector in the the direction of P M → \displaystyle \overrightarrow{PM} is ( 0 , 0 , 1 ) → \displaystyle \overrightarrow{(0, 0, 1)} . And we have F M → = F H → + H M → = ( 0 , 0 , 1 ) → + ( x_2d , y_2d , 0 ) → = ( x_2d , y_2d , 1 ) → \displaystyle \overrightarrow{FM} = \overrightarrow{FH} + \overrightarrow{HM} = \overrightarrow{(0, 0, 1)} + \overrightarrow{(\text{x\_2d}, \text{y\_2d}, 0)} = \overrightarrow{(\text{x\_2d}, \text{y\_2d}, 1)} .
Let P N → \displaystyle \overrightarrow{PN} be the angle bisector of F P → \displaystyle \overrightarrow{FP} and P M → \displaystyle \overrightarrow{PM} . And we have the direction of P N → \displaystyle \overrightarrow{PN} is that ( x_norm , y_norm , z_norm ) → + ( 0 , 0 , 1 ) → = ( x_norm , y_norm , z_norm + 1 ) → \displaystyle \overrightarrow{(\text{x\_norm}, \text{y\_norm}, \text{z\_norm})} + \overrightarrow{(0, 0, 1)} = \overrightarrow{(\text{x\_norm}, \text{y\_norm}, \text{z\_norm} + 1)} .
By the definition of the parabola, we have ∣ P M → ∣ = ∣ P F → ∣ \displaystyle | \overrightarrow{PM} | = | \overrightarrow{PF} | and thus △ P M F \displaystyle \triangle PMF is the isosceles triangle. This means that ∠ N P M = ∠ F M P \displaystyle \angle NPM = \angle FMP . Hence, we have P N → ∥ F M → \displaystyle \overrightarrow{PN} \parallel \overrightarrow{FM} . This means that P N → = λ F M → ⇒ ( x_norm , y_norm , z_norm + 1 ) → = λ ( x_2d , y_2d , 1 ) → ⇒ ( x_2d , y_2d ) = 1 z_norm + 1 ( x_norm , y_norm ) \displaystyle \overrightarrow{PN} = \lambda \displaystyle \overrightarrow{FM} \Rightarrow \overrightarrow{(\text{x\_norm}, \text{y\_norm}, \text{z\_norm} + 1)} = \lambda \overrightarrow{(\text{x\_2d}, \text{y\_2d}, 1)} \Rightarrow (\text{x\_2d}, \text{y\_2d}) = \frac{1}{\text{z\_norm} + 1} (\text{x\_norm}, \text{y\_norm}) .

// z < 0

Texel Solid Angle

By "10.4.4 Other Projections" of Real-Time Rendering Fourth Edition, the texel of the dual paraboloid map is more uniform than the cube map. By "20.4.1 Mapping and Distortion" of [Colbert 2007], the solid angle subtended by the texel of the dual paraboloid map is d ω = π ( ∣ z ∣ + 1 ) 2 \displaystyle d\omega = \pi{(|z| + 1)}^2 .

% here is the MATLAB code which verifies the meaning of "d_omega = (abs(z) + 1)*(abs(z) + 1)".

% "13.6.2 Sampling a Unit Disk" of [PBR Book](https://pbr-book.org/).
xi_1_size = 4096.0;
xi_2_size = 4096.0;
[ xi_1, xi_2 ] = meshgrid(linspace(0.0, 1.0, xi_1_size), linspace(0.0, 1.0, xi_2_size));
r = sqrt(xi_1);
theta = 2.0 .* pi .* xi_2;
u = r .* cos(theta);
v = r .* sin(theta);

% paraboloid
x = u;
y = v;
z = 0.5 - 0.5 .* (x .* x + y .* y);

% sphere
length = sqrt(x .* x + y .* y + z .* z);
x = x ./ length;
y = y ./ length;
z = z ./ length;

% "20.4.1 Mapping and Distortion" of [GPU Gems 3](https://developer.nvidia.com/gpugems/gpugems3).
d_omega = pi .* (z + 1.0) .* (z + 1.0);

% "numerical_omega" and "groundtruth_omega" are expected to be the same.
sum_numerator = sum(sum(d_omega));
sum_denominator = xi_1_size .* xi_2_size;
numerical_omega = sum_numerator ./ sum_denominator;

groundtruth_omega = 2.0 .* pi;

% output: "numerical:6.283569 groundtruth:6.283185".
printf("numerical:%f groundtruth:%f\n", numerical_omega, groundtruth_omega);

% here is the MATLAB code which verifies the meaning of "fWt = 4/(sqrt(fTmp)*fTmp)".

% "13.3.2 The Rejection Method" of [PBR Book](https://pbr-book.org/)
u_size = 4096.0;
v_size = 4096.0;
[ u, v ] = meshgrid(linspace(-1.0, 1.0, u_size), linspace(-1.0, 1.0, v_size));
r_2 = u .* u + v.* v;
chi = cast (r_2 < 1.0, class (r_2));

% paraboloid
x = u;
y = v;
z = 0.5 - 0.5 .* (x .* x + y .* y);

% sphere
length = sqrt(x .* x + y .* y + z .* z);
x = x ./ length;
y = y ./ length;
z = z ./ length;

% "20.4.1 Mapping and Distortion" of [GPU Gems 3](https://developer.nvidia.com/gpugems/gpugems3)  
d_omega = pi * (z + 1.0) .* (z + 1.0);

% "numerical_omega" and "groundtruth_omega" are expected to be the same.
sum_numerator = sum(sum(chi .* d_omega));
sum_denominator = sum(sum(chi .* 1.0));
numerical_omega = sum_numerator ./ sum_denominator;

groundtruth_omega = 2.0 .* pi;

% output: "numerical:6.283235 groundtruth:6.283185".
printf("numerical:%f groundtruth:%f\n", numerical_omega, groundtruth_omega);

Perspective-Correct Interpolation




References

[Heidrich 1998] Wolfgang Heidrich, Hans-Peter Seidel. "View-independent Environment Maps." EGSR 1998.
[Colbert 2007] Mark Colbert, Jaroslav Krivanek. "GPU-Based Importance Sampling." GPU Gems 3.
[Imagination 2017] Imagination. "Dual Paraboloid Environment Mapping." Power SDK Whitepaper 2017.