How to determine whether a point lies on Quad2DCurve
I have successfully drawn a Quad2D or Bezier curve in java. I have the equation for the same. But I need to determine whether a particular point(x,y) lies on the curve or not. I tried using Quad2D.contains and a few GeneralPath APIs, I could not 开发者_开发问答get the result accurately.
Can someone help to find out the solution to this?
I think you've meant QuadCurve2D class, which is quadratic Bezier curve.
There seems to be no ready-made method for that, and the problem comes to the distance from the point to the Bezier curve.
Let's P0 will be your point, P1
is a start point, P2
- a control point and P3
is an end point of your curve.
Then point on the curve will be given by
P = B(t)
There is such t
, for which distance between P
and P0
will be minimal.
F(t) = (B(t)_x - P0_x)^2 + (B(t)_y - P0_y)^2 -> min
If distance is 0 or less than certain error, then P0 is on the curve. t can be found with Netwon's iterative method by minimizing cost function
t_n = t_n-1 + F'(t) / F''(t)
where F'
is first derivative of cost function and F''
is its second derivative.
F'(t) = 2 * (B(t)_x - P0_x) * B'(t)_x + 2 * (B(t)_y - P0_y) * B'(t)_y
F''(t) = 2 * B'(t)_x * B'(t)_x + 2 * (B(t)_x - P0_x) * B''(t)_x +
2 * B'(t)_y * B'(t)_y + 2 * (B(t)_y - P0_x) * B''(t)_y
First derivative of quadratic Bezier curve B'(t) is a line segment with a start point (P2 - P1)
and end point (P3 - P2)
. Second derivative B''(t)
is a point P3 - 2 * P2 + P1
.
Plugging everything together will give a formula to t
for which F(t)
is minimal.
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