Project 1b

Points: 45 (+10 BONUS)

Purpose

Interact with smooth curves in OpenGL.

Set Up

Place \(N=8\) control points equally distributed to form a circle.
Title the window "yourFirstname yourLastname (ufid)"

For each Task below show the control points and the curve (sequence of line segments).
The points \(P_i\) are the same for all Tasks.

Task 1: Connect the points

Points: 5

Connect all the control points with white lines

In order to draw a closed polygon and draw line segments in white (instead of the point color), create a Vertex array of size ncpoints + 1. Copy the coordinate from cpoints, set the color to white and make sure the curve is closed.
Set up the VAO, VBO for this new array and call glDrawArrays with GL_LINE_STRIP to draw the polygon.

Task 2: (B-spline) Subdivision

Points: 10

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

Initialize \(P_i^0 = P_i\) (green).
Double the number of control points by setting

\[P^k_{2i} := \frac{P^{k-1}_{i-2} + 10P^{k-1}_{i-1} + 5P^{k-1}_{i}}{16}\] \[P^k_{2i+1} := \frac{5P^{k-1}_{i-1} + 10P^{k-1}_{i} +P^{k-1}_{i+1}}{16}\]

where \(k\) is the level of subdivison and
\(i\) is the index of points is in range \(0 \dots (N\times 2^k-1)\) .

Cycle through a maximal level of subdivisions: when \(k=0\) show the original control polygon. When key 1 is pressed and \(k=m\) then compute and draw 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: \(C^3\) Bezier

Points: 10

Another way to generate a curve is to construct several smoothly connected Bezier pieces.

We are going to construct 8 pieces of degree 4. The \(i\)th piece has control points \(\mathbf{c}_i=\{c_{i,0},c_{i,1},c_{i,2},c_{i,3},c_{i,4}\}\). One of the 5 control points is given as: \[c_{i,2} = \frac{4P_{i-1}+16P_i+4P_{i+1}}{24}\] and the points \(c_{i,1}\) and \(c_{i_3}\) have the following form: \[c_{i,1} = a P_{i-1} + b P_i + c P_{i+1}\] \[c_{i,3} = c P_{i-1} + b P_i + a P_{i+1}\] for constants \(a,b,c\).

Determine \(c_{i,0}\), \(c_{i,1}\), \(c_{i,3}\) and \(c_{i,4}\) so that the \(i\)th and the \((i+1)\)th Bezier piece join \(C^3\) . Write down the formulas for the control points into a readme.txt file.

Task 4: Draw the Bezier curve

Points: 15

For each Bezier curve piece:

Draw the 5 BB-coefficients in red

Choose \(n\) the number of pieces to discretize each Bezier piece into (tessellation number) sufficiently high so the discretized curve looks smooth.

For each Bezier curve piece, evaluate the points on the curve when \(t = 0, \frac{1}{n}, \frac{2}{n}, \dots , 1\) using De Casteljau's algorithm and connect the evaluation points with a polyline.
When 2 is pressed, show/hide the Bezier curves.

Task 5: Similarity between the curves

Points: 5

Compare smoothness and shape of the curves in Task 2 and Task 4 at maximal resolution.
Assume the 8 points are uniformly destributed on the circle. Is the resulting curve (evaluated at infinite resolution) a circle?
Justify your answer in the readme.txt file.

BONUS

Points: 10

Implement Task 4 using the OpenGL 4.x tessellation engine.

REMARK

Point penalty: 10

Make sure picking still works on the original \(N\) vertices, and your curves adapt to their movement.

WHAT TO SUBMIT

Your modified source files (.cpp's, shaders, etc)
A link to a screen capture of your running program showcasing the implementation of all of the tasks