Computing the minimum distance between two Bézier curves

Abstract

a b s t r a c tA sweeping sphere clipping method is presented for computing the minimum distance between two Bézier curves. The sweeping sphere is constructed by rolling a sphere with its center point along a curve. The initial radius of the sweeping sphere can be set as the minimum distance between an end point and the other curve. The nearest point on a curve must be contained in the sweeping sphere along the other curve, and all of the parts outside the sweeping sphere can be eliminated. A simple sufficient condition when the nearest point is one of the two end points of a curve is provided, which turns the curve/curve case into a point/curve case and leads to higher efficiency. Examples are shown to illustrate efficiency and robustness of the new method.

Computing the minimum distance between a point and a curve on mesh

“…The computation problem of the minimum distance between a point and a curve is also called the point projection problem of the curve [1][2][3][4][5][6]. The distance information between a point and a curve is very important for interactively selecting curves and curve construction in geometric modeling, for collision detection and physical simulation in computer graphics and computer vision, for interference avoidance in CAD/CAM and NC verification [4][5][6].…”

“…The distance information between a point and a curve is very important for interactively selecting curves and curve construction in geometric modeling, for collision detection and physical simulation in computer graphics and computer vision, for interference avoidance in CAD/CAM and NC verification [4][5][6]. However, most of the methods about this problem investigate the distance problem in the space R 3 , and only few methods study the distance problem on a curved space.…”

Hybrid Second-Order Iterative Algorithm for Orthogonal Projection onto a Parametric Surface

“…Curve F 1 (t) and F 2 (t) actually denote two surfaces in three-dimensional space, which are determined by the first and the second Formula in Equation (23), respectively. γ is the intersection point of two lines, the first of which is the intersection line of F 1 (t) and plane t, and the second of which is the intersection line of F 2 (t) and plane t. In another way, γ is the solution to Equation (23). Through point t 0 , draw a line perpendicular to the plane t, and then the perpendicular line intersects the surface F 1 (t) and F 2 (t), respectively.…”

“…Let γ = (α, β) T be a root of system F(t) = 0, t = (u, v) T , where the system F(t) = 0 is expressed by Formula (23). Define e n = t n − γ and use Taylor's expansion around γ, we have…”

“…Hu et al (2005) [8] use curvature iteration information for solving the projection problem. Scholars (Ku-Jin Kim (2003) [20], Li et al (2004) [21], Chen et al (2010) [22], [23], Bharath Ram Sundar et al (2014) [24]) have fully analyzed and discussed the intersection curve between two surfaces, the minimum distance between two curves, the minimum distance between curve and surface and the minimum distance between two surfaces. By clipping circle technology, Chen et al (2008) [25] provide a method for computing the minimum distance between a point and a NURBS curve.…”