Project 2b ©Jorg Peters

Purpose

Interact with smooth curves.

Set Up

For each Task below show

• the points and
• the curve as a sequence of short line segments.

The B-spline control points \(P_i\) are the same for all three Tasks.

The \(c_{i,j}\) in Task 3,4 are called BB-coefficients

Task 1: Connect the points

• Connect adjacent control points \(P_i\) with white line segments
  (Hint: use GL_LINE_LOOP or ip = i+1; if i==N, ip = 0; or modulo or repeat the end point when setting up the VAO, VBO for the polygon.)

Task 2: Subdivision

From the polygon in Task 1, generate a smoother curve by subdivision.

• Initialize \(P_i^0 = \) \(P_i\)
• Double the number of control points by setting
  \[P^k_{2i} := \frac{3}{4}P^{k-1}_{i}+\frac{1}{4}P^{k-1}_{i+1}\] \[P^k_{2i+1} := \frac{1}{4}P^{k-1}_{i}+ \frac{3}{4}P^{k-1}_{i+1}\]

  where \(k\) is the level of subdivison and
  \(i\) is in \(0, \dots, (N\times 2^k-1)\) .
  For example, if \(k=1,i=7\), \(P^{0}_{7} = \begin{bmatrix} 4 \cr 0 \end{bmatrix} \), \(P^{0}_{0} = \begin{bmatrix} 4c \cr 4s \end{bmatrix} \) then \(P^{1}_{14} = \begin{bmatrix} 3+c \cr s \end{bmatrix} \), \(P^{1}_{15} = \begin{bmatrix} 1+3c \cr 3s \end{bmatrix} \).

• Always show the original control polygon \(k=0\)
• Cycle through a maximal level of subdivisions: when key 1 is pressed and \(k=m\) then compute and draw in cyan the control polygon at level \(k=m+1\).
• However, when \(m=5\), \(m\) is set to 0 and the original control polygon is drawn.

Task 3: \(C^1\) Bézier (BB-form)

Another way to generate a curve is to construct several smoothly connected pieces in Bernstein-Bézier (BB) form.

We construct N BB-pieces of degree 2. The \(i\)th piece has BB-coefficients \(\mathbf{c}_i=\{c_{i,0},c_{i,1},c_{i,2}\}\).
The interior BB-coefficient is:

Determine \(c_{i,0}\) and \(c_{i,2} = c_{i+1,0}\) so that the polynomial pieces join \(C^1\).
Write down the formulas for the BB-coefficients in terms of the B-spline control points in readme.txt

Task 4: Draw the Bezier curve

Let \(n\) be the tessellation number (= number of evaluations-1 = number of segments) per BB-piece.

• For each BB-piece, apply De Casteljau's algorithm at \(t = 0, \frac{1}{n}, \frac{2}{n}, \dots , 1\) and connect the evaluated points with a polyline.
• Choose \(n\) sufficiently high so that the polynline looks smooth.
• When 2 is pressed, show/hide the Bezier curves.

Task 5: Is it a circle?

• Assume the \(N\) points \(P_i\) are uniformly destributed on the circle. Proof or disproof: is the resulting Bezier curve an exact circle?
• Justify your answer in the readme.txt file.

BONUS

Implement Task 4 using the OpenGL 4.x's tessellation engine. (You will have to use the web for details -- or sign up for Advanced Graphics) Make a note in the readme.txt file.

PENALTY

Make sure picking still works on the original \(N\) vertices, and your curves adapt when moved.

WHAT TO SUBMIT

• Your modified source files (.cpp's, shaders, etc) and a readme.txt file
• A link to a screen capture of your running program showcasing the implementation of all of the tasks using recordit (Mac, Win) or similar software.