CURVATURE ANALYSIS TUTORIAL

Transcription

1 CURVATURE ANALYSIS TUTORIAL NICHOLAS HEATHCOTT JOHN FERGUSON This tutorial will provide tools for analyzing the curvature of your geometry, be it linear, planar, or a solid form. This means both the degree or intensity of a curve, as well as its continuity over a surface. This tutorial covers the basics of curvature analysis, including internal curve smoothness, smoothness between curves, tools for visual analysis, and gaussian curvature analysis. 1. UNDERSTANDING CURVATURE ANALYSIS There are 3 ways of thinking about curves: a curve s tangent, its curavture circle, and vcurvature graphing. A. Curve Tangent The tangent line is the best straight line approximation of the direction vector of the curve at a given point. B. Curvature Circle The curvature circle measures the radius by which the tangent vector rotates at the same point. C. Curvature Graph The curvature graph visualizes the reciprocal of the curvature circle along the curve. In essence, where the line is higher, the curvature circle at that point would be small, representing a tighter curve. To enable curvature graphing, simply select a curve and type the CurvatureGraph command.

2 2. INTERNAL CURVE SMOOTHNESS Curves of different degrees have different internal smoothness at the transition between curve spans. Below are three curves of varying degrees drawn with the same control points. The knots are marked with red points. You can easily change the degree of your cirve with the ChangeDegree command. I. 2nd Degree 2nd Degree curves have internally continuous tangency. This is identifiable in the curvature graph, as the steps go in straight lines which means the tangent direction stays the same across the transition. II. 3rd Degree 3rd Degree curves have continuous curvature, so the graph does not have sudden steps like a 2nd degree curve. instead, the curvature graph is a continuous line over points of transition. III. 5th Degree 5th Degree curves have a totally smooth graph. In addition, the rate of change in the curvature is very smooth.

3 3. SMOOTHNESS BETWEEN CURVES Because models usually consist of many connected curves or surfaces, it is also important to understand how to analyze the smoothness of transitions between curves. There are three common types of transitions: position (G0), Tangency (G1), and curvature (G2). I. Position (G0) The curve ends touch, but there is no meaningful relationship between the two curves. II. Tangency (G1) The tangent direction stays uniform across the transition, but the curvature circle radius may change abruptly. The direction of tangency is determined by the first and second point on each curve, continuing until a curve is no longer tangent. III. Curvature (G2) The radius of curvature is the same at the transition endpoints. In this type of transition, light reflections will not break at the transition. Generally speaking, the surfaces should appear as one smooth surface.

4 4. VISUALIZING SURFACE SMOOTHNESS A. Environmental Mapping Environment Map (Analyze drop down > Surface > Environment Map) applies a rendered reflective surface to the geometry that allows the user to identify any cases of discontinuity. Simply select an object and enter the command Emap. G2 G1 CONTINUOUS CURVATURE G0 MINOR CURVATURE MAJOR CURVATURE B. Zebra Visualization Zebra (Analyze drop down > Surface > Zebra) applies a pattern to the surface that allows for easy identification on non continuous curvature. The example below outlines this on G0, G1, and G2 geometry. G2 G1 CONTINUOUS CURVATURE G0 MINOR CURVATURE MAJOR CURVATURE

5 5. GAUSSIAN CURVATURE ANALYSIS A. Example: Taurus Gaussian Curvature Analysis (Analyze drop down > Surface > Curvature Analysis > drop down Gaussian) applies a gradient to the selected geometry that shows the transition between curvature types and the directionality. You can also select your geometry and enter the command CurvatureAnalysis. CONVEX / CONCAVE POSITIVE GAUSSIAN CURVE SADDLE NEGATIVE GAUSSIAN CURVE SINGLE CURVE ZERO GAUSSIAN CURVE