8 Surfaces and Surface Modeling 8 Surfaces and

8. Surfaces and Surface Modeling The form given above does not satisfy the boundary

8. Surfaces and Surface Modeling Coons surface can be formed by using ruled surfaces.

8. Surfaces and Surface Modeling The normal to a surface is another important analytical

8. Surfaces and Surface Modeling The Hermite bicubic surface can be written in terms

8. Surfaces and Surface Modeling P(u, v) equation can be further expressed as:

8. Surfaces and Surface Modeling • Advantages – Boundary curves are Hermite curves –

8. Surfaces and Surface Modeling Open and closed Bezier surface examples

8. Surfaces and Surface Modeling For n=m=3, the equivalent bicubic formulation of an open

8. Surfaces and Surface Modeling where [P] is an (n +1)×(m +1) matrix of

8. Surfaces and Surface Modeling A triangular Bezier patch is defined by: For example,

8. Surfaces and Surface Modeling For n=4, the triangular patch is defined as;

8. Surfaces and Surface Modeling SUBDIVISION CURVES AND SURFACES

8. Surfaces and Surface Modeling Meshes methods Catmull Clark method: use to form a

8. Surfaces and Surface Modeling Catmull Clark method

8. Surfaces and Surface Modeling Loop Subdivision

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8. Surfaces and Surface Modeling

8. Surfaces and Surface Modeling The form given above does not satisfy the boundary conditions as shown below. Here below is a corrrection surface With the application of correction surface;

8. Surfaces and Surface Modeling

8. Surfaces and Surface Modeling Coons surface can be formed by using ruled surfaces.

8. Surfaces and Surface Modeling

8. Surfaces and Surface Modeling The normal to a surface is another important analytical property. The surface normal at a point is a vector which is perpendicular to both tangent vectors at the point. And the unit normal vector is given by:

8. Surfaces and Surface Modeling The Hermite bicubic surface can be written in terms of the 16 input vectors: ; Hermite matrix ; geometri ya da sınır koşulu matrisi

8. Surfaces and Surface Modeling P(u, v) equation can be further expressed as:

8. Surfaces and Surface Modeling

8. Surfaces and Surface Modeling • Advantages – Boundary curves are Hermite curves – Interior points can be controlled • Disadvantages –What should be the twist factor? It is not esay to sense the effect of twist vector(Ferguson pacth twist vector is 0). – Cannot be used with high order polynomials.

8. Surfaces and Surface Modeling

8. Surfaces and Surface Modeling Open and closed Bezier surface examples

8. Surfaces and Surface Modeling

8. Surfaces and Surface Modeling For n=m=3, the equivalent bicubic formulation of an open and closed cubic B -spline surface can be derived as below.

8. Surfaces and Surface Modeling where [P] is an (n +1)×(m +1) matrix of the vertices of the characteristic polyhedron of the B-spline surface patch. For a 4× 4 cubic B-spline patch:

8. Surfaces and Surface Modeling

8. Surfaces and Surface Modeling A triangular Bezier patch is defined by: For example, a cubic triangular patch is;

8. Surfaces and Surface Modeling For n=4, the triangular patch is defined as;

8. Surfaces and Surface Modeling

8. Surfaces and Surface Modeling SUBDIVISION CURVES AND SURFACES

8. Surfaces and Surface Modeling

8. Surfaces and Surface Modeling Meshes methods Catmull Clark method: use to form a quadrilateral mesh. produces a smoother surface This method tends to move edge vertices at corners more than other outer vertices.

8. Surfaces and Surface Modeling Catmull Clark method

8. Surfaces and Surface Modeling Loop Subdivision

8. Surfaces and Surface Modeling