Spline – basically, a smooth curve with a constantly changing radius – discussion

Complex surface – surface with a constantly changing radius

If we consider the best approximate circle radius that passes through a point on a complex surface or spline curve, the reciprocal of the radius – 1/r – of this circle is the curvature of the surface or curve at that point.

If a surface is nearly planar – flat – then a point on it will have a very large approximate radius – 1/r will give a low curvature.

If a surface is has a very tight bend in it then a point on it will have a very small approximate radius – 1/r will give a high curvature.

In the above image:

The red line is the surface cross section or a spline curve

The yellow circles are the approximate radius at those points

The blue line with grey spines is the Curvature Plot – the longer the spine, the higher the curvature, small radius – high curvature, large radius – small curvature.

The curvature plot is applied to the convex side of the curve, the point where the curve flips from convex to concave is called the point of inflexion.

 

Arcs or Splines?

Where possible, always use splines for complex surface sections.  Multiple arcs will give multiple patches in the resultant quilt with only G1 endpoint/boundary relations.

Enlarge image

 

Section Curvature – Splines and Surfaces

Always start with the minimum amount of control in a curve or surface.  The system as to adhere to your specified control points/curve whether they are smooth or not.  Better to allow the system to find the smoothest path between minimum control points/sections.