Implicitization via Moving Curves

[Zorrent12]

Homework Statement

Hello. I am currently working with bezier curves in a programming project and have need to calculate the implicit equation of a cubic bezier curve. After doing some research, I decided to take the moving curve approach.

From what I can gather, the first step to implicitization of a cubic curve is finding a quadratic function from the following matrix.

| 3*P_0*P_1 3*P_0*P_2 P_0*P_3 | | (1 - t)^2 |
| 3*P_0*P_2 P_0*P_3 + 9*P_1*P_2 3*P_1*P_3 | | (1 - t)*t |
| P_0*P_3 3*P_1*P_3 3*P_2*P_3 | | t^2 |

Where each point is multiplied by taking the cross product of their coordinates. I chose the top function and defined the following variables

a00 = 3*(y0*1 - y1*1)
b00 = 3*(x1*1 - x0*1)
c00 = 3*(x0*y2 - x1*y0)

a01 = 3*(y0*1 - y2*1)
b01 = 3*(x2*1 - x0*1)
c01 = 3*(x0*y2 - x2*y0)

a02 = y0*1 - y3*1
b02 = x2*1 - x0*1
c02 = x0*y3 - x3*y0

representing the function as

(x*a00 + y*b00 + c00)*(1 - t)^2 +
(x*a01 + y*b01 + c01)*(1 - t)*t +
(x*a02 + y*b02 + c02)*t^2

The next step was to find a function of one degree less. I read this could be done by shifting the bottom row to the right and eliminating the bottom left corner. I defined the following additional variables

a10 = a02;
b10 = b02;
c10 = c02;

a11 = 3*(y1*1 - y3*1);
b11 = 3*(x3*1 - x1*1);
c11 = 3*(x1*y3 - x3*y1);

and representing the function as

(x*a10 + y*b10 + c10)*(1 - t) +
(x*a11 + y*b11 + c11)*t

With the two functions

(x*a10 + y*b10 + c10)*(1 - t) +
(x*a11 + y*b11 + c11)*t

and

(x*a00 + y*b00 + c00)*(1 - t)^2 +
(x*a01 + y*b01 + c01)*(1 - t)*t +
(x*a02 + y*b02 + c02)*t^2

I calculated the dot product between P(t) = (x, y, 1) and the functions. I then attempted to calculate the resultant using the Sylvester matrix, which should represent the implicit equation of the curve.

| (x*a00 + y*b00 + c00) (x*a01 + y*b01 + c01) (x*a02 + y*b02 + c02) |
| (x*a10 + y*b01 + c00) (x*a11 + y*b11 + c11) 0 |
| 0 (x*a10 + y*b01 + c00) (x*a11 + y*b11 + c11) |

(x*a00 + y*b00 + c00)*(x*a11 + y*b11 + c11)*(x*a11 + y*b11 + c11) +
(x*a01 + y*b01 + c01)*0*0 +
(x*a02 + y*b02 + c02)*(x*a10 + y*b01 + c00)*(x*a10 + y*b01 + c00) -
(x*a02 + y*b02 + c02)*(x*a11 + y*b11 + c11)*0 -
(x*a01 + y*b01 + c01)*(x*a10 + y*b01 + c00)*(x*a11 + y*b11 + c11) -
(x*a00 + y*b00 + c00)*0*(x*a10 + y*b01 + c00) = 0

The trouble is, when I plugged in the coordinates of a point on the curve, the implicit equation resulted in a number other than zero.

Homework Equations

Cross Product: (a, b, c) x (d, e, f) = (bf - ec, dc - af, ae - db)

Determinant of a 3x3 matrix:
a_11*a_22*a_33 +
a_12*a_23*a_31 +
a_13*a_21*a_32 -
a_13*a_22*a_31 -
a_12*a_21*a_33 -
a_11*a_23*a_32

The Attempt at a Solution

For my test, I used a curve made up of the control points

P_0 = 2, 2
P_1 = 1, 3
P_2 = 3, 4
P_3 = 5, 5

And chose to test the equation using the two endpoints. The first point resulted in the equation equaling 648.0, while the other resulted in -3240.0.

Thank you for taking the time to read my lengthy post. I apologize if the information provided is inadequate. If there is need for more clarification, please ask.

 

[mighty2000]

Hello, it seems like you have made some progress in your approach to calculating the implicit equation of a cubic bezier curve. However, I noticed a few issues that may be causing your results to be incorrect.

First, it seems like you may have made a mistake in defining your additional variables a10, b10, and c10. These variables should be based on the bottom row of the original matrix, not the bottom left corner. So a10 should be 3*(y2*1 - y3*1), b10 should be 3*(x3*1 - x2*1), and c10 should be 3*(x2*y3 - x3*y2). Similarly, a11, b11, and c11 should be based on the bottom row of the shifted matrix, not the bottom left corner. So a11 should be 3*(y0*1 - y2*1), b11 should be 3*(x2*1 - x0*1), and c11 should be 3*(x0*y2 - x2*y0).

Second, it seems like you may have made a mistake in calculating the dot product between P(t) and the functions. The dot product should be between the vector (x, y, 1) and the functions, not between P(t) and the functions. So the dot product should be (x*a00 + y*b00 + c00)*(1 - t)^2 + (x*a01 + y*b01 + c01)*(1 - t)*t + (x*a02 + y*b02 + c02)*t^2 and (x*a10 + y*b10 + c10)*(1 - t) + (x*a11 + y*b11 + c11)*t.

Lastly, it seems like you may have made a mistake in calculating the determinant of the Sylvester matrix. The determinant should be a_11*a_22*a_33 + a_12*a_23*a_31 + a_13*a_21*a_32 - a_13*a_22*a_31 - a_12*a_21*a_33 - a_11*a_23*a_32, not (x*a00 + y*b00 + c00)*(x*a11 + y*b11 + c11)*(x*a11 + y*b11 + c11) + (x*a01 + y*b01 + c01