Big picture: You will create an artistic/fun pretzel-like 2d B-spline curve. This curve will be used to  create a CONS (on a Bezier surface). You will make two more copies of the CONS, moving them in 3D space, so that the resulting three 3d curves form a chain, and then you will create a 3d print of this object. Steps for this are described below.

Objectives:

-- experience designing with B-splines

-- more surface design experience

-- improve understanding of 3d geometry

-- understand the concept of Curves on Surfaces (ConS)

-- hands-on experience with 3d printing

Details:

step 1) Using  one 2d cubic B-spline curve, design an artistic curve with a pretzel-like shape (at least two loops) in the xy-plane.

The B-spline must have the following properties.

-- at least ten polynomial segments,

-- full multiplicity at the ends,

-- start point is equal to the end point (closed but not periodic)

-- at least one interior domain knot must have multiplicity 2

-- at least one interior domain knot must have multiplicity 3

The control points should live in [0,1] x [0,1] in order to live in the domain of a Bezier surface. (If needed, you can use a translation and/or scale to achieve this.)

You may use MM’s B-spline function BSplineCurve[] for display, but you will need your own evalua- tor to extract (x,y) points (that will be interpreted as (u,v) for the surface).  Your own evaluator will     use BSplineBasis[]. (You may use code from the demo nbs.)

Plot your curve in 2D with the control points and polygon, using different colors for the curve and control polygon/points.

Print the knot vector and the number of curve segments next to this plot.

step 2) Design an interesting/curvey Bezier patch. The patch must be at least bicubic. (You could use the Coons method.)

step 3) Plot your B-spline curve as a ConS together with the surface. Include the Bezier control net in this plot.

step 4) Plot the ConS without the surface displayed. Use the Tube function to give it thickness.    Please take some time to set the thickness parameter to a “reasonable” value because this will    influence the 3D print quality. (Too thin and the print will not be stable enough; too thick and your design concept is lost.)

step 5) Translate the ConS twice, creating three curves that are chained together and do not inter- sect.

This will require a bit of “eyeballing it.” You could modify your ParametricPlot3D from step 4 to look