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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The surface normal at a point is a vector which is perpendicular to both tangent vectors at the point Figure 2 ; that is. A parametric surface patch with its boundary conditions The twist vector at a point on a surface measures the twist in the surface at the point.

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: Write down the bilinear surface equation using Equation 4.

## L11 Hermite Bicubic Surface Patch

The principle curvatures are the roots of the following equation: Estimate the tangent vectors using the 4 corner points. And the unit normal vector is given by. 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 hemrite. Bilinear Surface A bilinear surface is derived by interpolating four data points, using linear equations in the parameters u and v so that hsrmite resulting surface has the four points at its corners, denoted P 00P 10P 01and P 11as shown in Figure 1.

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.

A parametric surface patch with its boundary conditions. EFand G are the first fundamental, or metric, coefficients of the surface.

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The tangent vectors at the corner points can be approximated in terms of the corner surfzce using the direction and the length of chord lines joining the corner points. surcace

Thus, the boundary matrix for the F-surface patch becomes Figure 3. Similarly, Surfacf uv is obtained from P 0 v and P 1 v as.

A bilinear surface Deriving the equation of a bilinear surface would be equivalent to finding the expression of the coordinates of an arbitrary point corresponding to the parameter values u and v.

Because the blending functions are linear, the bilinear surface tends to be flat. Write a simple Matlab program to draw the surface. 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 surfface v1- u vuv. 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.

We assume that this point is obtained by dividing the line segment between P 0 v and P 1 v in the ratio u: The Ferguson surface also called the F-surface patch is a herrmite surface patch with zero twist vectors at the patch cornersas shown in Figure bicubjc. Do the results make sense to you? Thus, the boundary matrix for the F-surface patch becomes. Check out the definition of a?

The Gaussian curvature at a point on the surface is defined as. Geometric properties of surfaces 1. 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.

For a 2-D parametric curve represented in terms of its arc length s. This is the course surfce for? The mean curvature 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.

Deriving the equation of a bilinear surface would be equivalent to finding the expression of the coordinates of an arbitrary point corresponding to the parameter values u and v. This special surface is useful in design and machining applications. Describe how you can determine the shortest distance between, say, San Francisco and Taipei.

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Use proper mesh and bifubic to display the surface. Show your Matlab program too.

Define 4 corner points in 3D space. 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 material is be used strictly for teaching and learning of this course. 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.

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 surfaace Equation 4. Substituting Equation 1 and 2 into Equation 3 gives the bicbic equation of a bilinear surface: Based on these assumptions, P 0 v and P 1 v are.