Advanced Methods in Computer Graphics by Ramakrishnan Mukundan

Author:Ramakrishnan Mukundan
Language: eng
Format: epub
Publisher: Springer London, London

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The tangent vectors m 1 and m 2 can have arbitrary magnitude if we require only G 1 continuity at the end points when two curves are joined together. Increasing the magnitude causes the curve to align closer to the tangent direction. A scale parameter α > 0 for the tangents is introduced into this equation to control the shape of the cubic curve:

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α is sometimes referred to as the tension parameter of the curve. An example with four different values of α is shown in Fig. 7.4b. Note that when α = 0, the above equation represents a linear interpolation between P 1 and P 2.

Given n points (n > 2), we can develop an interpolating spline that passes through all the points by constructing Hermite cubic curves for every consecutive pair of points. The tangent direction at each knot must be carefully specified by the user in such a way that it corresponds to the tangents to curves on both sides of the knot.

In Fig. 7.5, piecewise Hermite polynomial curves are fitted through a set of points. The points are the same as the knots of the interpolation curve shown in Fig. 7.2. The common tangent vectors are all defined as parallel to negative y-axis.

Fig. 7.5Hermite interpolation spline

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