When it comes to Nurbs vs Bezier, the CAD forums are rife with arguments and counter arguments. The following is an excerpt of my post in one of those forums. It describes some of the affecting factors.

### Approximation

Digital representation of shape is based on approximation. There are many different algorithms and equations which tries to describe a shape to a certain tolerance. Take a circle for example. If we were to fit a circular shape with Beziers, we would end up with a multitude of Bezier segments (of reasonable degree.. 3, 5, 7 .. definitely not 20) connected to one another, and still the radius values along the segment might vary very slightly. A common problem while make wheels.. Using Nurbs ( with different weights) of degree 2 we can describe a circle with far less information (CVs knots. etc). The shape is precise with radius values same all along the circle. ( the locus information might still be different.. too difficult to explain using only text).There is a miss-conception that NURBS means just multi span. The power of nurbs comes from the fact that the CVs can have different influence(weights, rationality) on the geometry. In short nurbs can describe a particular shape with the least amount of data, as compared to any other approximation.

What is shape?

Shape is what can be described verbally to describe an object. Vocabulary of shapes are pretty limited. Some examples are spherical, cylindrical, cuboid, flat, round, convex, concave, ring, torus, conical. Complex design can be made by combining two or more shapes in a legible manner, so as to create different feeling. If one notices carefully the shapes can be defined by relatively simple mathematical equations or functions. Let us call those surfaces the Slabs surfaces or Base surfaces. If we were to list down the characteristics of the base surfaces, they would be as follows.

Introduction to Digital Sculpting
Digital Sculpting is sculpting 3d form using computer programs. By sculpting we mean interactive and free form manipulation of form.

Different types
Can be broadly classified into Surface based and Polygon based. Surface based sculpting is mainly used when accuracy is of paramount importance, such as when used for manufacture of Cars, white goods, toys to name a few. Polygon based sculpting is used for Animation, Visualization, Computer Games etc. A 3d form can be described by a contiguous set of polygons, by triangulation. So polygonal models can be automatically generated from 3d scan data or 3d CMM data.

Rhinoceros started as a application to run within AutoCAD using AGLib from Applied Geometry. Alias was the largest user of AGLib. The following are the excerpts from the Rhino Wiki site.

• May 92 - First meeting with AG. Applied Geometry (AG) came to Mcneel for assistance in integrating their AGLib, NURBs geometry library, in AutoCAD. AG customers included Alias Research, Spatial, Honda, and Tecnomatix
• Jul 92 - First prototype in AutoCAD after about 3 days of work.
• Nov 92 - McNeel/AG agreement to develop AccuModel, NURBs modeling for AutoCAD. McNeel would do the marketing and AG would do all the development. McNeel would provide AutoCAD development support as needed.
• Nov 92 - Michael Gibson hired as an intern. He brought Sculptura, a mesh modeler that he had done as a class project.
• Mar 93 - Sculptura released.
• Mar 93 - McNeel takes over as the lead on the AccuModel development.
• Jul 93 - First prototype of Sculptura 2 ready for NURBs.
• Nov 93 - Sculptura 2 nicknamed RhinoCeros.
• Jan 94 - New McNeel/AG agreement where McNeel was licensing AGLib from AG and AG would provide the needed AGLib enhancements and maintenace.
• Apr 94 - Rhino beta released on the Graphic Alternative BBS. This was our first version of a public beta program.
• May 94 - Dr Dale Lear hired from AG. We had found that we needed geometry expertise in house in order to develop the functionality and usability needed by our clients.
• May 94 - Alias Research agrees to purchase AG. Alias was AGs largest customer and they felt that one of their advantages was the geometry technology.
• May 94 - First commercial products completed using AccuModel. Ed Monk & Son Navel Architect released an 82 foot sports fishing boat.
• Jul 94 - Geometry development begins in earnest. Alias didnt seem to be interested in the geometry library business.
• Aug 94 - Sculptura renamed RhinoCeros after it was determined that Mcneel would not be able to reslove the trademark problems with AccuModel.
• Aug 94 - Private showing of Rhino at SIGGRAPH.
• Nov 94 - Sale of AG to Alias final.
• Mar 95 - McNeel provides Alias with first installment of geometry technology.
• Jun 95 - Alias purchased by Silicon Graphics.
• Jun 95 - McNeel receives the last update of AGLib.
• May 97 - Last build of AccuModel for AutoCAD. As the Rhino product progressed we decided that it would be better to focus on the Windows version.
• Sep 97 - 50,000 beta sites and growing fast without any promotional effort on our part.
• Jul 98 - 100,000 beta sites.
• Jul 98 - Announced Oct release at SIGGRAPH.
• Oct 98 - Release Rhino version 1.0.
• Dec 98 - First 5,000 shipped.
• Jan 99 - Japanese version released.
• Jan 99 - First public beta of 1.1 released.
• Jan 99 - First European reseller meeting in Barcelona.
• Mar 99 - First third party book on Rhino published.
• May 99 - Korean version ships.
• Aug 00 - First public beta of 2.0 and Flamingo released.
• Sep 01 - Shipping Rhino 2.0.
• Nov 01 - Shipping Flamingo.
• Nov 01 - Inside RhinoCeros published
• Jan 03 - Shipping Rhino 3.0
• Jan 03 - Shipping Flamingo 1.1
• Jul 03 - Shipping Penguin 1.0
• Jul 04 - Shipping Bongo 1.0
• Jan 07 - Shipping Rhino 4.0
• Jan 08 - Shipping Penguin 2.0
• Jul 08 - Shipping Flamingo 2.0
• Feb 09 - Shipping Brazil for Rhino

Follows us

While modifying or correcting an existing model, be prepared to throw away the model what you have painstakingly made. Modelers often try to hang on to and salvage what they have already made, while adding new features and modifications. Many a times, starting afresh improves the quality of the model and will be faster. That would mean re-making/modifying the basic/slab surfaces and theoretical surfaces.

Follows us