A planar cubic Bézier spiral
Traditionally, the smooth connection between different curves (e.g., the connection between straight and curved portions of a railway track where continuous curvature transition is of the essence) has been done by transcendental curves. In order to apply CAD methods to construction problems of the aforementioned type, it would be useful to have Bézier curves to replace clothoid curves and circular involutes. The authors give the needed tool in constructing a spiral arc (i.e., an arc of monotone curvature) given by a starting point \({\mathbf B}\), starting and ending tangent vectors \({\mathbf T}\), \({\mathbf T}_1\), making an angle \(\theta\), and a final curvature value \(c\). Define the two parameters \(a=25\tan\theta/54c\cos\theta\), \(b=5\tan\theta/9c\). Then the cubic Bézier curve \[ {\mathbf Q}(u)= {\mathbf B}(1- u)^3+3{\mathbf B}_1(1-u)^2u+ 3{\mathbf B}_2(1-u) u^2+{\mathbf B}_3 u^3,\quad 0\leq u\leq1, \] where \({\mathbf B}_1={\mathbf B}+a{\mathbf T}\), \({\mathbf B}_2={\mathbf B}_1+a{\mathbf T}\), \({\mathbf B}_3={\mathbf B}_2+b{\mathbf T}_1\), is a spiral arc having curvature \(0\) at \({\mathbf B}\) \((u=0)m\), curvature \(c\) at \({\mathbf B}_3\), and tangent vectors \({\mathbf T}\), \({\mathbf T}_1\) at the endpoints. The authors work out all the needed formulas for connections of straight line to circle, circle to circle in all cases of relative positions, and straight line to straight line. This is a very useful paper.
scientific article; zbMATH DE number 933155