homework #1

due: Mar 22 2006

credits: 100

description

-Write an OpenGL program to implement a Spline Sketcher.

[5] Draw a 500×500 window with black background. Set the title of the window to "grade_id / name". (See Fig.1)

[10] Place six control points P₀, P₁, P₂, P₃, P₄, P₅ in the window as follows: control points P₀, P₁, P₂, P₃, P₄, P₅equally distributed on the unit circle

grade_id / name

P₂-----P₁ / \ / \ P₃ P₀ \ / \ / P₄-----P₅

Draw the control points P_i with the size of 5×5 pixels in green and connect them with white lines to form the control polygon.

Plot the spline curves defined by the control polygon above. We will have 5 curve types:
1. [10] When the key '1' is pressed,
  for each i = 0, ..., 5, (suppose i = i mod 6)
  - cⁱ₀ := ( P_i + P_i-1 ) / 2
  - cⁱ₁ := P_i
  - cⁱ₂ := cⁱ⁺¹₀
  Draw the points cⁱ_j in cyan with the size of 3×3 pixels.
  
  Use the function glMap1() to draw the quadratic curves with their control points of cⁱ₀, cⁱ₁, cⁱ₂. (See Fig.1)
  When the "C" key is pressed draw a circle of appropriate size to show that the piecewise polynomial curve is not a circle.
2. [5] When the key '2' is pressed,
  
  for each i = 0, ..., 5, (suppose i = i mod 6)
  - cⁱ₁ := ( 2*P_i + P_i+1 ) / 3
  - cⁱ₂ := ( P_i + 2*P_i+1 ) / 3
  Determine cⁱ₀ and cⁱ₃ so that the polynomial pieces join C².
  
  Draw those points in red color with the size of 3×3 then connect them to form the piecewise linear curves.
  
  Use the function glMap1() to draw the cubic curves with their control points of cⁱ₀, cⁱ₁, cⁱ₂, cⁱ₃.(See Fig.2)
3. [15] Determine rational weights for quadratic and cubic so that the resulting curve, when rendering using glMap1(), is a circle. When key 'q' is pressed,
  show the circle for quadratic and when key 'c' is pressed, show the circle for cubic. (See Fig.q for quadratic and Fig.c for cubic)
4. [10] When the key '3' is pressed,
  use a single glNurbsCurve() function call to plot the cubic curve and the quadratic curve (See Fig.3). Rotate and superimpose the rotated curves to see they are not circular.
5. [15] To approximate the length of the curve, compute the length of the control polygon.
  Each time when 's' key is pressed, get a better approximation by performing one step of de Boor's subdivision (see note2 and 6.14 in papers2).
  
  Redraw the new subdivided control polygon and print out the new length (in the command window, not the OpenGL window).
Note:

Pick different colors (as you like) to draw the curves.
Fig.1 Fig.2 Fig.3

Fig.q Fig.c

[15] Allow the position of control points to be interactively modified: when the left mouse button is clicked sufficiently near to a control point, that point will be picked and follow the motion of mouse cursor. Refresh the control polygon and the curve while the mouse is moving. Releasing the left button stops the motion.
[15] Allow the resizing of the window. Adjust the control points and the curve according to the size of the window. The picking and dragging of the control points should work as above regardless of the size of the window.
Program exits when ESC key is pressed.

demo program

Here are the executables for each platform you can try yourselves. In the demo you can hide the curves by pressing space bar, but this is not required in your program.


Fig.1	Fig.2	Fig.3

Fig.q	Fig.c