What we want is a vertex normal, a normal at each point making up the grid that is perpendicular to the Bézier surface. But this technique only produces a face normal.įaceNormal = (v1 - v0).crossProduct(v2 - v0) įigure 2: a normal at any point on the patch can be computed from a cross product between the tangent on the surface at that point along the \(u\) and the \(v\) direction. We can compute the cross-product between two edges making up a face. How can we generate a normal at the position of each vertex of the grid? We can easily generate a face normal using a technique similar to the one we have used in the getSurfaceProperties method of the TriangleMesh class. Generating the position of the points is something we can already do for shading, but we also need a normal. Now that we studied a couple of techniques to create a Bézier patch, a problem remains. Figure 1: a face normal can be computed from the cross product between two edges of a face. Rendering Curves as Geometry: Hair RenderingĬalculating Normals of Bézier Surfaces Reading time: 7 mins.It is visible that the properties of end-points on Q-Bézier curve and surface can be locally controlled by these shape parameters. The Q-Bézier curve and surface inherit the most properties of Bézier curve and surface and can be more approximated to the control polygon.
It has been shown that the main advantage compared to the ordinary Bézier curves and surfaces is that after inputting a set of control points and values of newly introduced n shape parameters, the desired curve or surface can be flexibly chosen from a set of curves or surfaces which differ either locally or globally by suitably modifying the values of the shape parameters, when the control polygon is maintained. The corresponding Bézier curves and surfaces, the so-called Quasi-Bézier (i.e., Q-Bézier, for short) curves and surfaces, are also constructed and their properties studied. It is a natural extension to classical Bernstein basis functions. Firstly, a class of polynomial basis functions with n adjustable shape parameters is present. It is a novel generalization of Bézier curves and surfaces. A new formulation for the representation and designing of curves and surfaces is presented.
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