Abstract
The Bishop frame or rotation minimizing frame (RMF) is an alternative approach to define a moving frame that is well defined even when the curve has vanished second derivative, and it has been widely used in the areas of computer graphics, engineering, and biology. The main aim of this paper is using the RMF for classification of singularity type of timelike sweeping surface and Bishop spherical Darboux image which is mightily concerning a unit speed spacelike curve with timelike binormal vector in E 1 3 .
Highlights
A sweeping surface is a surface traced by a oneparameter family of spheres with centers on a regular space curve, its directrix or spine
A new frame is needed for the kind of mathematical analysis that is typically done with computer graphics
Bishop [17] introduced the rotation minimizing frame (RMF) or Bishop frame, which could provide the desired means to ride along a space curve with vanished second 1derivative
Summary
A sweeping surface is a surface traced by a oneparameter family of spheres with centers on a regular space curve, its directrix or spine. There are several examples that we are familiar with, such as circular cylinder (spine is a line, the axis of the cylinder), right circular cone (spine is a line (the axis), radii of the spheres not constant), torus (directrix is a circle), and rotation surface (spine is a line). This visualization is a popularization of the classical notation of a partner of a planar curve [1,2,3,4].
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