计算机视课件13.pptVIP

Course 13  Curves and Surfaces * * Course 13 Curves and Surface ----- Surface Representation Representation Interpolation Approximation ----- Surface Segmentation Segmentation Registration Basic Geometry 1)?Curve: Parameter form: e.g., a line segment between and : ? Implicit form: Explicit form: 2)?Surface: Parameter form: Implicit form: Explicit form: (graph surface) 3) Planes: Assume are 3 non-collinear points in space. A plane is uniquely determined by the points. Let The equation of the plane is where is the normal of the plane. In parametric form of the plane, we can redefine Then the plane equation is: where are parameters, 4) Surface Curvatures: Assume a surface in form of The normal of the surface of point is where are partial derivatives of the surface. (1)??Normal curvature: The intersection of a surface and a plane that passes through the normal at a point P of the surface is a 2D curve. The curvature of the 2D curve at point P is called normal curvature of the surface at point P. ---- At a surface point. The surface has infinite many normal curvatures. ---- The minimum and maximum normal curvatures, K1 and K2 are called principal curvatures. (2) Gaussian and mean curvatures Gaussian curvature: Mean curvature: (3) Types of a surface region K(x,y) ? ? + 0 – ? ? ? + peak ridge saddle ridge 0 ? flat minimal surface – pit valley saddle Valley (4) Curvature Calculation For a surface 2. Representation of curve and surface 1)?Curve representations: ----- Piecewise curve segments -----Joint condition: smooth at least in the first order derivatives,i.e. continuous in curve slope. ----- Lowerest order of polynomials. This leads to cubic spline curve: x(t) = axt3 + bxt2 + cxt + dx y(t) = ayt3 + byt2 + cyt + dy z(t) = azt3 + bzt2 + czt + dz When one uses: then a c