Two families of cubic Hermite curves forming a parametric net are the basis of the bicubic Hermite surface. Learn more about Chapter 7: Bicubic Hermite. Parametric Bicubic Surfaces. ▫ The goal is to go from curves in space to curved surfaces in space. ▫ To do this, we will parameterize a surface. bicubic surfaces. We want to define We want to define smooth surfaces too. Parametric but also have the nice Hermite property of continuous tangent vector.
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Construct a sphere to represent the earth. Use 4 control points to generate a Bezier curve in 2D space. We can verify that the four data points are at the corners of the bilinear surface hermitw substituting the proper combination of 0 and 1 for u and v in Equation 4.
Use proper mesh and view to display the surface. A bickbic surface is derived by interpolating four data points, using linear equations in the parameters u and v so that the resulting surface has the four points at its corners, denoted P 00P 10P 01and P 11as shown in Figure 1.
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L11 Hermite Bicubic Surface Patch – Free Download PDF
Bilinear Surface A bilinear surface is derived by interpolating four data points, using linear equations in the parameters u and v so that the resulting surface has the four points at its corners, denoted P 00P 10P 01and P 11as shown in Figure 1. Hence, the designer does not have to input tangent vector information and the computations required to calculate the surface parameters are simplified.
Equation 11 can be further expressed as. Define 4 corner points in 3D space. Equation 17 is often called the first fundamental quadratic form of a surface and is written as. The mean curvature is defined as. Thus, the boundary matrix for the F-surface patch becomes. Equation 4 also tells us that a bilinear surface is obtained by simply blending the effects of the corner points after they are weighted by the blending functions 1- u 1- vu 1- v1- u vuv. The Ferguson surface also called the F-surface patch is a bicubic surface patch with zero twist vectors at the patch cornersas shown in Figure 3.
L11 Hermite Bicubic Surface Patch
Based on these assumptions, P 0 v and P 1 v are. For a 2-D parametric curve represented in terms of its arc length s. As shown in Figure 2, 16 boundary conditions are required to find blcubic coefficients of the equation. Show your Matlab program too. Construct the boundary yermite matrix [B] in Equation The point P uv defined in this way will traverse the entire surface consistently with the increment of the parameter values u and v from 0 to 1.
The tangent vectors at the corner points can be approximated in terms of the corner positions using the direction and the length of chord lines joining the corner points. The Gaussian curvature at a point on the surface is defined as. The principle curvatures are the roots of the following equation: Deriving the equation of a bilinear surface would be surdace to finding the expression of the coordinates of an arbitrary point corresponding to the parameter values u and v.
A parametric surface patch with its boundary conditions. Finally, write blcubic the surface equations of your F-surface patch and write a simple Matlab program to draw the F-surface. This is useful if tool paths are to be generated to mill the surface. The attributes to be stored would be only four corner points because the surface represented by Equation 4 can be reconstructed from these four corner points.
Hermite Bicubic Surface is an extension of Hermitd cubic spline.
And the unit normal vector is given by. Bicubic surfaces provide designers with better surface design uermite when designing surfaces. Check out the definition of a? Thus, the boundary matrix for the F-surface patch becomes Figure 3.
Referring to Figure 5, when we rotate the plane around the normal, the curvature varies and has a maximum and minimum value in two perpendicular directions.
Draw the shortest path on the sphere. Do the results make sense to you? Substituting Equation 1 and 2 into Equation 3 gives the following equation of a bilinear surface: Check out the latitudes and longitudes of Taipei, Tokyo, the North Pole, and San Francisco, and mark approximately the positions of the 4 cities on the earth. The bicubic equation can be written as. Principle Curvature The principle curvatures are the roots of the following equation: Write down the bilinear surface equation using Equation 4.
The paths that have minimum lengths are analogous to a straight line connecting two points in euclidean space and are known as geodesics. In general, two distinct points on a surface can be connected by many different paths, of different lengths, on the surface.
We assume that this point is obtained by dividing the line segment between P 0 v and P 1 v in the ratio u: For a 2-D parametric curvenote thatandEquation 21 can be transformed into.
We also assume that P 0 v and P 1 v are the internal division points of the line segments P 00 P 01 and P 10 P 11 respectively, in the ratio v: The surface normal at a point is a vector which is perpendicular to both tangent vectors at the point Figure 2 ; that is.
Use proper mesh and views to display the surface. The normal to a surface is bicbic important analytical property. The Hermite bicubic surface can be written in terms of the 16 input vectors: