Re: Tracking rectangles and bezier paths
Re: Tracking rectangles and bezier paths
- Subject: Re: Tracking rectangles and bezier paths
- From: Wagner Truppel <email@hidden>
- Date: Sat, 9 Sep 2006 17:51:53 -0700
Ok, then I have two other suggestions...
a) Upon checking the documentation for NSBezierPath <http://
developer.apple.com/documentation/Cocoa/Reference/ApplicationKit/
Classes/NSBezierPath_Class/NSBezierPath_Class.pdf>, I learned of the
method -containsPoint:, which "Returns a Boolean value indicating
whether the receiver contains the specified point". Maybe that's all
you need, that is, check whether the path contains the mouse point
and, if so, redraw the path with a thicker stroke and/or a different
color, so as to highlight the path. The documentation is actually not
very clear as to whether it's testing for a point being *on* the path
or only inside the path's region, so you'll have to try and see.
b) If that's not enough, you can use NSBezierPath's -bounds method to
retrieve the bounding rectangle of the path itself (which is never
larger than the convex hull of the path's control points, and often
smaller) and use that as a fast first-pass on deciding whether or not
the mouse is near the path. If the mouse is inside the path's bounds
rectangle, then you can retrieve the path's control points and do
some math of your own to find out if the mouse point is on the path
itself, as follows.
Reading the documentation <http://developer.apple.com/documentation/
Cocoa/Conceptual/CocoaDrawingGuide/CocoaDrawingGuide.pdf>, I learned
that NSBezierPath creates *cubic* Bezier curves, that is, curves
whose points satisfy the parametric equation
P(t) = (1-t)^3 P0 + 3t(1-t)^2 P1 + 3(1-t)t^2 P2 + t^3 P3
where t is a real number in the closed interval [0,1] and P0, P1, P2,
and P3 are the control points (P0 and P3 are the starting and ending
points of the curve, whereas P1 and P2 are true control points in the
sense that they're not on the curve itself). Wikipedia has an
excellent intro to Bezier curves, by the way, if you want to learn
more <http://en.wikipedia.org/wiki/Bézier_curve>.
So, once you know that the mouse point is near the path (namely,
inside its bounding rectangle), you can retrieve the control points,
set P(t) to the mouse point, and solve for t. If you find a solution
in the range [0,1], then the mouse point is *on* the path.
Now, solving for t is a little tricky, because it involves solving a
polynomial equation of 3rd order (for which a general solution exists
but is cumbersome). Moreover, since the P's are two-dimensional
entities (points), you need to solve the cubic using only the points'
x coordinates (or only their y coordinates), then use the value of t
found (if you find one in the interval [0,1], that is) to test
whether the parametric equation above is satisfied for the other
component. That's more efficient than solving for t twice (once for
the x coordinates and once for the y coordinates) and then comparing
the two solutions for equality.
To actually solve for t, expand the powers in P(t) above and collect
terms of identical powers in t to arrive at something like this
A t^3 + B t^2 + C t + D = 0
where A, B, C, and D are expressions depending on the various control
points and on the mouse point, but not on t, and then use the method
outlined in, for example, <http://www.karlscalculus.org/pdf/
cubicQuartic.pdf#search="solving cubic polynomials"> to solve
for t (using either the x or the y components, as I pointed out above).
Hope this helps.
Wagner
============
The difference between an auto mechanic and a quantum mechanic is
that the quantum mechanic can get the car inside the garage without
opening the door.
Yes, that is correct. I am interested in highlighting the curve
when the mouse is on it. Apple does this very nicely in Space
Designer (part of Logic Pro). I would be interested to know how
they do this. Anyone from Apple care to comment?
Dave
On Sep 9, 2006, at 11:10 AM, Raphael Sebbe wrote:
Hi,
From what Dave told, I believe he is interested in highlighting the
curve when the mouse is *on* it, as opposed to *inside* the area
defined by it. The convex hull may be too large for tracking purpose.
Raphael
On 9/9/06, Wagner Truppel <email@hidden> wrote:
It's not as hard a math problem as suggested, even if the curve is
not monotonic. Recall that the entire bezier path is contained
within
the convex hull of the path's control points so a very simple
solution is to retrieve the control points, compute their convex
hull
(which is a convex polygon, in 2 dimensions), and use that as your
tracking area. Computing convex hulls in 2 dimensions is almost
trivial - any decent book on algorithms will tell you how.
Wagner
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