VARKON Version 1.15 1997-10-16

### The Conic Lofting surface

A major problem in surface modeling is often that the number of constraints you impose on the shape of a surface is larger than the mathematical representation of the surface really accepts. A simple bicubic patch for example is uniquely defined by 48 numbers and nothing more. If this amount of freedom is not enough to describe a certain shape you can either use more patches or you can raise the order of the basic polynomial from 3 to 5 or even higher or you can use rational polynomials.
None of these solutions are particularly good however. Defining a surface with many patches requires more data and increases the risk of introducing oscillations or other errors. Raising the order of the polynomial is a well documented way of introducing oscillations and unstable numerical behavior. Rational polynomials are better then non rational but the difference is relatively small.
The VARKON Conic Lofting surface was developed with many years of bad experience from current modeling techniques in mind and is therefore based on a completely new concept. The representation of LFT_SUR is analytically defined by a section of parametric rational cubic segments in the V-direction and by a procedural method in the U-direction. Each conic in the section is controlled by 2 limit curves, 2 tangent curves and a midpoint curve or P-value curve (MP). The 6:th curve is called the spine.
sur_conic(id,spine,lim1,tan1,mpflag1,MP1,lim2,tan2.......limn,tann);
To evaluate the surface VARKON first uses the parametric U-value to calculate a corresponding position on the spine and then sets up a plane with the spine tangent as normal. The plane is then used to intersect the 5 curves that define the conic and use this data to create the conic which in the end is evaluated for the parametric V-value. The result is a surface that has the characteristics of the conic in the V-direction and the characteristics of the limit and tangent curves in the U-direction. The spine controls the parametrisation in the V-direction.
Surfaces based on LFT_SUR are automatically smooth in the U-direction since a conic can not oscillate. The quality of the surface in the V-direction depends on the shape of the limiting curves but is independent of their parametrisation. This gives the LFT_SUR a large amount of built in smoothness. LFT_SUR also has a unique degree of freedom to be formed with a minimum of input data. A CUB_SUR may need 100 patches or more to approximate the shape of one LFT_SUR patch within acceptable limits.
Evaluation of a LFT_SUR can require heavy computations but saves a lot of memory compared to traditional surfaces.

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