Transforming implicit surface (distance field) and their gradient
Transform a 3D field function / distance field and its gradient with a 4x4 matrix - 10/2016 - #Distance Field
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Let f:\mathbb R^3 \rightarrow\mathbb R be any distance field. We can transform any implicit surface f(x,y,z) = c with a 4x4 transformation matrix as follows:
\hat f(\vec p) = f( \mathbf T^{-1} \vec p)
Gradient
Sometimes you need to compute the normal of an implicit surface. You usually do this using the gradient \nabla f which needs to be transformed as well. Similar to the normal of a mesh use the inverse transpose of \mathbf T :
\hat{ \nabla f}(\vec p) = \left ( \mathbf T^{-1} \right )^{T} . \nabla f ( \mathbf T^{-1} \vec p )
Code
float transformed_value_and_gradient(Point pos, Vec3& grad)
{
Point p = T.inverse() * pos;
float field_values = distance_field(p, grad); // original field function value
grad = T.inverse().transpose() * grad;
return field_values;
}
Special transformations
Special space transformations such as bending, twisting, warping, wobbling, duplicating, mirroring are also possible and discussed at length in other articles
- Inigo Quilez provides a long list of manipulations (end of article)
- Ronja's tutorials also some other space manipulations
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