Minimum-Area Ellipse With Three Non-Collinear Points And Its Application In Parametric Interpolation

In the field of Computer Graphics and Computer-Aided Geometric Design, modeling the shape of an image or an object usually requires the technique of parametric curve interpolation, i.e., constructing a parametric curve that interpolates, or passes through, a given set of data points. There are a lot of methods to define and express a curve. Parameter spline curve is one of the most widely used methods. Usually, the constructed parametric interpolation curve is required to be smooth. Even with the same interpolation method, if the data point parametric method is different, the shape of the obtained curve is different. The data points should be selected in a way so that good approximation is guaranteed. Generally speaking, we hope the curve parameterization will be as closer to the arc length parameterization as possible.Presently, there are several popular methods in use for choosing knots location, namely, uniform parameterization method, accumulated chord length method, centripetal method, Foley’s method, and ZCM method. The uniform method is applicable to the occasion that each side (or chord) of the data points polygon is nearly equal. When the chord length of the adjacent data points varies widely, the uniform method will make the parametric curve bad. The accumulated chord length method, which seems to be the most widely used one, directly reflects the distribution of the data points in accordance with the chord length. When the data points achieved is dense enough, the chord length can be seen as the approximation of arc length, interpolation curve will be better smooth. The centripetal method and Foley’s method are the adjustment and optimization of the chord length method. ZCM method determines knots in the construction of the parametric interpolating spline curves, and obtains a reproduction degree of two.In theory, the arc length parametric is optimal parametric method, namely the points on the curve is proportional to the corresponding point in the parameters domain. We know that the arc length of parametric curve is obtained by the integral equation, in general, which has no analytic expression, so that the accurate implementation of arc length parametric is impossible.The problem of determining a minimum-area ellipse through three non-collinear points is discussed in this paper. We give the proof and construction of the minimum-area ellipse through three non-collinear points from the geometric point of view, and present a new method of determining knots. This method replaces the chord length, which is closer to the arc length of the minimum-area ellipse with arc length, and avoids the occurrence of ’oscillation’ and ’loops’. We compare the new method with the uniform method, chord length method and the centripetal method. The comparison is performed on the quality of cubic spline curves using these methods. In most cases, the result of our method is better than others.