|length and energy of quadratic bézier curves and applications|
|Young Jo on Ahn a, Christoph Hoﬀmann b and Paul Rosen|
ABSTRAK: This pap er derives expressions for the arc length and the b ending energy of quadratic B ezier curves. The formulae are in terms of the control p oint co ordinates. For xed start and end p oints of the B ezier curve, the lo cus of the middle control p oint is analyzed for curves of xed arc length or b ending energy. In the case of arc length this lo cus is convex. For b ending energy it is not. Given a line or a circle and xed end p oints, the lo cus of the middle control p oint is determined for those curves that are tangent to the given line or circle. For line tangency, this lo cus is a parallel line. In the case of the circle, the lo cus can b e classi ed into one of six ma jor typ es. In some of these cases, the lo cus contains circular arcs. These results are then used to implement fast algorithms that construct quadratic B ezier curves tangent to a given line or circle, with given end p oints, that minimize b ending energy or arc length.
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