Quadratic Birational Maps And The Construction Of PH Transition Curve

Image morphing technology is the basis for the development of computer graphics and digital image technology, which is widely used on television, advertising, medicine. Image transformation operation involves mapping problem between different regions. Typically, people do not consider the inverse mapping. However, in order to facilitate the subsequent application of deformation technology, we need to study the inverse mapping. In particular, if the map is a bi-rational mapping. That is, the mapping and its inverse mapping are both rational. Computing based on rational map is convenient and efficient. On the other hand, by the classical theory of Cremona Transform, any bi-rational mapping of higher order (degree>2) can be expressed by the composition of several quadratic bi-rational maps. Thus it is important to study the construction of quadratic bi-rational maps. Given control points, how to get quadratic bi-rational map by attaching more freedom to every control points is major work.At present, T. W. Sederberg presented a method to construct bi-rational map of bi-degree (1,1). The specific method is to attach a weight to each control point. When these weights satisfy one special equation, the mapping is a bi-rational map. However, it is a kind of special quadratic bi-rational map. In order to construct general quadratic bi-rational map, we use the technique of moving lines. Thus we get a system of algebraic equations. Solving these equations, we can get general bi-rational map. At the same time, these equation can be converted to a uni-variate polynomial equation of degree four, so we can get the closed form of the solutions.A cubic PH curve used to construct the transition curve between two circles (in the case of that one circle is not inside the other), which satisfies the G2 Continuous condition. Due to the fact that one circle is not inside the other, the curvature of the curve at the endpoint is the same sign, so they are able to construct C-shaped transition curve. In certain case, we can prove the existence of only cubic PH transition curve between two circles and give the transition curve construction algorithm.