• Advantages of Bezier Curves:
  • Derivative information is not required
  • Users can have an intuitive feel
• Properties of Bezier Curves:
  • Only the first and last vertices lie on the curve

• The Binomial Coefficient (\(n\) choose \(i\)):
  • \(\binom{n}{i} = \frac{n!}{i!(n-i)!}\)
• The Basis Function:
  • \(J_{n,i}(t) = \binom{n}{i} t^i (1-t)^{n-1}\)
  • where \(n\) is the degree of the polynomial and \(i\) is the vertex index (in \([0,n]\))
• Points on the Curve:
  • \(P(t) = \sum_{i=0}^{n} P_i J_{n,i}(t)\) for \(t \in [0,1]\)
  • where \(P_i\) is the vector containing the coordinates of vertex \(i\)

• Evaluators:
  • Compute the values for Bernstein polynomials of any order
• Types:
  • Points/vertices are the most common (e.g., GL_MAP1_VERTEX_3)
  • Can also be used for colors, normals, and textures
  • Must be enabled using glEnable() (e.g., glEnable(GL_MAP1_VERTEX_3))
• Arguments:
  • umin and umax define the range of the parameter (usually 0 and 1)
  • stride is the number of values of the parameter between curve segments
  • order is the order of the polynomial plus 1

svaexamples/curves/draw.c (Fragment: curve)

   //
   // Create and draw the curves for every 4 points
   for (int i=0; (i+3)<numPoints; i+=3)
   {
      // The evaluator with a stride of 3 and an order of 4
      glMap1f(GL_MAP1_VERTEX_3, 0.0, 1.0, 3, 4, &points[i][0]);

      // Draw the curve
      glBegin(GL_POINTS);  // GL_LINE_STRIP fills the gaps
      {
         for (int i=0; i<100; i++)
         {
            glEvalCoord1f(((float)i)/100.0);             
         }
      }
      glEnd();
   }
        

svaexamples/curves/draw.c (Fragment: onMouseClick)

/**
 * The mouse callback (i.e., the function that is called
 * each time a mouse button is pressed or released).
 *
 * @param button The button (e.g., GLUT_LEFT_BUTTON)
 * @param state  The state (e.g., GLUT_UP or GLUT_DOWN)
 * @param x      The x-position of the mouse
 * @param y      The y-position of the mouse
 */
void onMouseClick(int button, int state, GLfloat x, GLfloat y)
{
   if(button == SVA_LEFT_BUTTON)
   {
      if (state == SVA_DOWN)
      {
         points[numPoints][0] = x;
         points[numPoints][1] = y;
         points[numPoints][2] = 0.;

         numPoints++;         
         
         svaRedisplay();         
      }
   }
   else if (button == SVA_RIGHT_BUTTON)
   {
      if (state == SVA_DOWN)
      {
         selectedPoint = closestPointTo(x, y, 0.0);         
      }
      else
      {
         selectedPoint = -1;         
      }

      svaRedisplay();         
   }
}
        

svaexamples/curves/draw.c (Fragment: onMouseDrag)

/**
 * The mouse drag callback
 *
 * @param x      The x-position of the mouse
 * @param y      The y-position of the mouse
 */
void onMouseDrag(GLfloat x, GLfloat y)
{
   if (selectedPoint >= 0)
   {
      points[selectedPoint][0] = x;
      points[selectedPoint][1] = y;

      svaRedisplay();         
   }
}
        

svaexamples/curves/draw.c (Fragment: onDisplay)

/**
 * Display Callback
 */
void onDisplay()
{
   // Clear the screen
   glClear(GL_COLOR_BUFFER_BIT);

   // Set the color
   glColor3f(1.0, 0.0, 0.0);

   //
   // Create and draw the curves for every 4 points
   for (int i=0; (i+3)<numPoints; i+=3)
   {
      // The evaluator with a stride of 3 and an order of 4
      glMap1f(GL_MAP1_VERTEX_3, 0.0, 1.0, 3, 4, &points[i][0]);

      // Draw the curve
      glBegin(GL_POINTS);  // GL_LINE_STRIP fills the gaps
      {
         for (int i=0; i<100; i++)
         {
            glEvalCoord1f(((float)i)/100.0);             
         }
      }
      glEnd();
   }
   
         
   // Instead of:
   //
   //   glBegin(GL_POINTS);
   //   {
   //     for (int i=0; i<100; i++)
   //     {
   //       glEvalCoord1f(((float)i)/100.0);             
   //     }
   //   }
   //   glEnd();
   //
   // we could use:
   //
   //   glMapGrid1f(100, 0.0, 1.0 );         
   //   glEvalMesh1(GL_POINT, 0, 100);

       
    // Show the points
   glPointSize(3.);
   glBegin(GL_POINTS);
   {
      for (int i = 0; i<numPoints; i++)
      { 
         if (i == selectedPoint)  
         {
            glColor3f(1.0, 0.0, 1.0);
            glVertex3f(points[i][0], points[i][1], points[i][2]);   
         }
         else
         {
            glColor3f(0.0, 1.0, 0.0);
            glVertex3f(points[i][0], points[i][1], points[i][2]);   
         }
      }
      glEnd();
   }
   

   // Connect the points (using a piecewise linear curve)
   if (numPoints >= 2)
   {
      glColor3f(0.75, 0.75, 0.75);
      glBegin(GL_LINE_STRIP);
      {
         for (int i = 0; i < numPoints; i++)
         { 
            glVertex3f(points[i][0], points[i][1], points[i][2]);        
         }        
      }
      glEnd(); 
   }
}