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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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. 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. We assume that this point is obtained by dividing the line segment between P 0 v and P 1 v in the ratio u: 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 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.
Based on these assumptions, P 0 v and P 1 v are. Similarly, P uv is obtained from P 0 v and P 1 v as. Substituting Equation 1 and 2 into Equation 3 gives the following equation of a bilinear surface: 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.
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 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.
Define 4 corner points in 3D space. Write down the bilinear surface equation using Equation 4. Write a simple Matlab program to draw the surface.
Use proper mesh and views to display the surface. Show your Matlab program too. Because the blending functions are linear, the bilinear surface tends to be flat. Bicubic surfaces provide designers with better surface design tools when designing surfaces. The bicubic equation can be written as. Hermite Bicubic Surface is an extension of Hermite cubic spline.
As shown in Figure 2, 16 boundary conditions are required to find the coefficients of the equation.
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. 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.
The normal to a surface is another important analytical property. It is used to calculate cutter offsets for three-dimensional NC programming to machine surfaces, volume calculations, and shading of a surface model.
The surface normal at a point is a vector which is perpendicular to both tangent vectors surrace the point Figure 2 ; that is. And the unit normal vector is given by. The Hermite bicubic surface can be written in terms of the 16 input vectors: Equation 11 can be further expressed as.
L11 Hermite Bicubic Surface Patch
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 sufface. Thus, the boundary matrix for the F-surface patch becomes.
This special surface is useful in design and machining applications. 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.
Hence, the designer does not have to input tangent vector information and the computations required to calculate the surface parameters are simplified. This bkcubic useful if tool paths are to be generated to mill the surface.
Estimate the tangent vectors using the 4 corner points. Construct the boundary bicubiv matrix [B] in Equation Finally, write down the surface equations of your F-surface patch and write a simple Matlab program to draw the F-surface. Use proper mesh and view to display the surface. In general, two distinct points on a surface can be connected by many different paths, of different lengths, on the surface. The paths that have minimum lengths are analogous to a straight line connecting two points in euclidean space and are known as geodesics.
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. Construct a sphere to represent the earth. 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. Check out the definition of a? Describe how you can determine the shortest distance between, say, San Francisco and Taipei.
Draw the shortest path on the sphere. Equation 17 is often called the first fundamental quadratic form of a surface and is written as. EFand G are the first fundamental, or metric, coefficients of the surface. For a 2-D parametric curvenote thatandEquation 21 can be transformed into.
For a 2-D parametric curve represented in terms of its arc length s. Use 4 control points to generate a Bezier curve in 2D space. Do the results make sense to you?
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. We call the extreme values the? The principle curvatures are the roots of the following equation: The Gaussian curvature at a point on the surface is defined as. The mean curvature is defined as.
L11 Hermite Bicubic Surface Patch – Free Download PDF
This is the course material for? This material is be used strictly for teaching and learning of this course. Geometric properties of surfaces 1. 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.
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. Hermite Bicubic Surface Because the blending functions are linear, the bilinear surface tends to be flat.
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. Thus, the boundary matrix for the F-surface patch becomes Figure 3. Geodesics In general, two distinct points on a surface can be connected by many different paths, of different suurface, on the surface. Principle Curvature The principle curvatures are the roots of the following equation: