In , computer researcher Martin Newell needed a new 3D model for his work. In these days of age, very few models where available to the computer graphics community and creating them was also far from easy. Most models had to get their points entered in the computer program by hand or with a graphics tablet "a computer input device that allows hand-drawn images and graphics to be input. It may be used to trace an image from a piece of paper laid on the surface.

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Dragging the small triangle i. This point is referred to as the tracing point. This system can also show the detailed computations step-by-step.

To see this, select Window followed by Triangular Computing Scheme. This will bring up the Triangular Computing Scheme Window. Clicking on STEP displays the left-most column which contains all given control points as show below: Clicking on STEP again displays the second column, each element of which is computed as the combination of the north-west and south-west control points on the first column.

The south-east pointing and north-east pointing arrows have values 1-u and u, respectively. Note that the points on each column correspond to a polyline of the de Casteljau net. Since we have two columns, we have two polylines of the de Casteljau net. Clicking on STEP twice produces the following result. As you can see, we have four columns and hence four polylines of the de Casteljau net.

Clicking on STEP twice again will produce a single point, which is the point on the curve corresponding to the given u. The vertical and horizontal sliders in the Triangular Computing Scheme Window are used for scrolling the triangular-shaped computation vertically and horizontally, respectively. Since its degree is 5, only six control points are involved. Clicking on STEP will display the first column which is actually part of the control polygon. On-the-Fly Computation One of the many nice features the curve system provides to you is that you can drag the u-indicator to trace the curve and at the same time see the change of convex hull, de Casteljau or de Boor net, and the triangular computation scheme for obtaining the corresponding point on the curve.

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## De Casteljau's Algorithm Revisited

When has a value of 0, you will get point. When has a value of 1, you will get point. The next simplest version of a Bezier curve is a quadratic curve, which has a degree of 2 and control points. A quadratic curve is just a linear interpolation between two curves of degree 1 aka linear curves. Specifically, you take a linear interpolation between , and a linear interpolation between , and then take a linear interpolation between those two results. That will give you your quadratic curve.

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## Paul de Casteljau

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