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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Use 4 control points to generate a Bezier curve in 2D space.

Hermite Bicubic Surface is an extension of Hermite cubic spline. 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. Check out the definition of a? Use proper mesh and views to display the surface.

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 mean curvature is defined as.

As shown in Figure 2, 16 boundary conditions are required to find the coefficients of the equation. EFand G are the first fundamental, or metric, coefficients of surfacw surface. Principle Curvature The principle curvatures are the roots of the following equation: The twist vector at a point on a surface measures the twist in the surface at the point.

Do the results make sense to you? Because the blending functions are linear, the bilinear surface tends to be flat. 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: Referring to Figure 5, when surfac rotate the plane around the normal, the curvature varies and has a maximum and minimum value in two perpendicular directions.

A parametric surface patch with its boundary conditions. This material is be used strictly for teaching and learning of this course.

Write a simple Matlab program to draw the 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. For a 2-D parametric curvenote thatandEquation 21 can be transformed into. We can verify that the four data points are at the corners of the bilinear surface by substituting the proper combination of 0 and 1 for u and v in Equation 4. Bicubic surfaces provide designers with better surface design tools when designing surfaces.

### L11 Hermite Bicubic Surface Patch – Free Download PDF

Hence, the designer does not have to input tangent vector information and the computations required to calculate the surface parameters are simplified. The normal surfzce a surface is another important analytical property. In general, two distinct points on a surface can be connected by many different paths, of different lengths, on the surface.

The Hermite bicubic surface can be written in terms of the 16 input vectors: This is the course material for? Show your Matlab program too. Write down the bilinear surface equation using Equation 4.

## L11 Hermite Bicubic Surface Patch

Surface geodesics can, for example, provide optimized motion planning across a curved surface for numerical control machining, robot programming, and winding of coils around a rotor.

They are the 4 corner data points, the 8 tangent vectors at the corner points two at each point in the u and v directionsand the 4 twist vectors at the corner points. This special surface is useful in design and machining applications. 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.

Substituting Equation 1 and 2 into Equation 3 gives the following equation of a bilinear surface: The Gaussian curvature at a point on the surface is defined as.

The paths that have minimum lengths are analogous to a straight line connecting two points in euclidean space and are known as geodesics. This is useful if tool paths are to be generated to mill the surface. Draw the shortest path on the sphere. 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.