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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Draw the shortest path on the sphere. 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.
The attributes to hetmite stored would be only four corner points because the surface represented by Equation 4 can be reconstructed from these four corner points. Geometric properties of surfaces 1.
And the unit normal vector is given by. Use 4 control points to generate a Bezier curve in 2D space. 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 bicubicc. Define 4 corner points in 3D space.
Write down the bilinear surface equation using Equation 4. Thus, the boundary matrix for the F-surface patch becomes Figure 3. This is useful if tool paths are to be generated to mill the surface. The Gaussian curvature at a point on the surface is defined as.
In general, two distinct points on a surface can be connected by many different paths, of different lengths, on the surface. 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.
Similarly, P uv is obtained from P 0 v and P 1 v as. 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. Hermite Bicubic Surface Because the blending functions are linear, the bilinear surface tends to be flat.
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. 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.
Sueface 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.
Finally, write down the surface equations of your F-surface patch and write a simple Matlab program to draw the F-surface.
Thus, the boundary matrix for the F-surface patch becomes. Equation 11 can be further expressed as. Construct the bicuubic condition matrix [B] in Equation Use proper mesh and view to display the surface.
L11 Hermite Bicubic Surface Patch – Free Download PDF
EFand G are the first fundamental, or metric, coefficients of the surface. Write a simple Matlab program to draw the surface. A bilinear surface is derived by interpolating four data points, using linear equations in bicubid 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.
Construct a sphere to represent the earth.
L11 Hermite Bicubic Surface Patch
The bicubic equation can be written as. Principle Curvature The principle curvatures are the roots of the following equation: Show your Matlab program too. 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. It is used to calculate cutter offsets for three-dimensional NC programming to machine surfaces, volume calculations, and shading of a surface hemrite.
Estimate the tangent vectors using the surfae corner points. Because the blending functions are linear, the bilinear surface tends to be flat. We call the extreme values the? Bilinear Surface A bilinear surface is derived by interpolating four data points, using linear equations in the parameters u and v herite that the resulting surface has the four points at its corners, denoted P 00P 10 surgace, P 01and P 11as shown in Figure 1.
Bicubic eurface provide designers with better surface design tools when designing surfaces. The surface normal at a point is a vector which is perpendicular to both tangent vectors at the point Figure 2 ; that is. Equation 17 is often called the first fundamental quadratic form of a surface and is written as.
Geodesics In general, two distinct points on a surface can be connected by many different paths, of different lengths, on the surface. This special surface surrace useful in design and machining applications.
Hence, the designer does not have to input tangent vector information and the computations required to calculate the surface parameters are simplified. For a 2-D parametric curvenote thatandEquation 21 can be transformed into.
A parametric surface patch with its boundary conditions The twist vector at a point surfsce a surface measures the twist in the surface at the point. Substituting Equation 1 and 2 into Equation 3 gives the following equation of a bilinear surface: As shown in Figure 2, 16 boundary conditions are required to find the coefficients of the equation.
The principle curvatures are the roots of the following equation: