I am trying to write a program to draw a bezier curve segment implementing the knowledge from my textbook. I wanted to try making a drawing program to draw something using implicit functions. I want my program to work as follow.
Theoretically, I can draw the curves by direct substitution, but I might do some modifications to the curve and I want to implement what I have learnt, so, I wanted to do it wisely(direct substitution seems dumb).Thz^^
Edit 1 : Let's assume the boundaries were provided by the user
The first step is generating a parameterized expression for the curve. The given example can be transformed very easily:
c(t) = (t, 2 * t^3)^T
Now express this curve in Monomial basis:
c(t) = / 0 1 0 0 \ * (1, t, t^2, t^3)^T
\ 0 0 0 2 /
= C * M(t)
In this expression, the first matrix C is the coefficient matrix. All we need to do is transform this matrix to Bernstein basis. The matrix that transforms Monomial basis to Bernstein basis is:
/ 1 - 3t + 3t^2 - t^3 \ / 1 -3 3 -1 \ / 1 \
B(t) = | 3t - 6t^2 + 3t^3 | = | 0 3 -6 3 | * | t |
| 3t^2 - 3t^3 | | 0 0 3 -3 | | t^2 |
\ t^3 / \ 0 0 0 1 / \ t^3 /
This equation can be inverted to get:
/ 1 1 1 1 \
M(t) = | 0 1/3 2/3 1 | * B(t)
| 0 0 1/3 1 |
\ 0 0 0 1 /
Substituting this into the curve equation, we get:
c(t) = C * M(t)
/ 1 1 1 1 \
= C * | 0 1/3 2/3 1 | * B(t)
| 0 0 1/3 1 |
\ 0 0 0 1 /
The first matrix product can be calculated:
c(t) = / 0 1/3 2/3 1 \ * B(t)
\ 0 0 0 2 /
And this gives you the control points for the Bezier curve:
p0 = (0, 0)^T
p1 = (1/3, 0)^T
p2 = (2/3, 0)^T
p3 = (1, 2)^T
This very procedure can be applied to any polynomial curve.
The general solution for an equation in the form
y = a + b * x + c * x^2 + d * x^3
is:
p0 = (0, a)^T
p1 = (1/3, a + b/3)^T
p2 = (2/3, a + 2b/3 + c/3)^T
p3 = (1, a + b + c + d)^T