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The curve fitting technique is somewhat involved and makes use of several different algorithms. That said, the sample code is complete and commented, so it can be used to fill in the details. The parameter, t, ranges from 0 to 1, and Q t outputs x,y coordinates on my bezier curve. The first thing to do is calculate estimated values of t that match up to the points on my path.


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That is, Q t[index] should equal the point on my path at index. After generating inputs for Q t , the next step is to try to fit one bezier curve to the all the points on the path given the inputs. Here points 1 and 4 are the end points and 2 and 3 are the control points. The orange lines just visually connect the control point with its corresponding end point.

The blue curve is the bezier itself. The direction of the control points relative to their end points is also simple to determine: simply compute the vector between the path end points and their nearest neighbor. In this case the direction of the left control point is the vector between points 1 and 2 in the path.

The right control point direction is the vector between points 5 and 6. The only unknown in constructing our bezier curve is the distance between the control points and their respective end point.

A Display Algorithm of Brush Strokes Using Bézier Functions

By controlling the distance of the control points from their end points, I can affect the curve of the bezier. I want to chose distances leftAlpha and rightAlpha such that it best fits the points in the path. Formally stated, I want to minimize the squared errors as represented by:. Where point[i] is the point on the path at index i, t[i] is one of the estimated parameters I generated earlier, and Q is bezier curve. Remember that point[i] should be equal to Q t[i] , assuming the bezier curve fits perfectly my points.

Right now though, I want to minimize S when calculating leftAlpha and rightAlpha. Using the least squares approach I can write the equations I want to solve as:. Here d is derivative, usually written as the Greek letter delta. S is the squared errors defined above. Where i ranges between [ Berstein[i] is a polynomial that is highly regarded as boring for this discussion. Look at the code if you really want to know. After a lot of mathematical voodoo and gymnastics see the paper for all of that , I arrive at:.

I know that t has to range from 0 to 1, and Q t is supposed to correspond to the points on my path.

Zbrush - applying stroke along curve

Q 0 should be the first point on my path, and Q 1 should be the last. The initial estimate of t uses the chord length method. Ideally, t would be the length of curve from the left end point to the point corresponding to t, divided by the total length of the curve. However, the chord length method makes the assumption that a straight line is a reasonable estimate of a curve. So to estimate t, I measure the length of the line segments from the left end point to the point corresponding to t, then divide that by the length of all the line segments.

When did this all start?

Here Q t —point is the vector from a point on the bezier curve to the corresponding point on my path. Although I skipped over a lot of detail when it came to curve fitting, hopefully this post has been helpful in understanding how it works, in addition to explaining the vector brush. For the details that I skipped over, the code should fill in the blanks. The vector brush is a fun tool to play with and the curve fitting algorithm has other applications too.

Shodo with a brush pen, "curve and hook" stroke, kanji sample, heart, spear, light

Safe from the Losing Fight. I create Mac and iOS software. Naive Implementation The direct way to implement the vector brush would be to simply record the mouse movements into a NSBezierPath. Brush strokes, simplified At this point the best way to improve the my brush is to remove all the redundant points from our NSBezierPath. I repeat the process on [1, 2, 3] as my path: Here my end points are 1 and 3, and my only interior point is 2.

Thus, the final result is: The Code As you might expect the code for this algorithm is rather simple. Curvy is better So now I have a much simplified path representing my brush stroke. Curve fitting overview The curve fitting technique is somewhat involved and makes use of several different algorithms. Fitting a bezier curve to points Remember that a bezier curve is defined by four points, like below. Calculating parameters i. Conclusion Although I skipped over a lot of detail when it came to curve fitting, hopefully this post has been helpful in understanding how it works, in addition to explaining the vector brush.

Then, from polygonal approximations of the fitted trajectory and the brush outline, a polygonal approximation of the stroke outline is computed.

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Last, the polygonal approximation of the stroke outline is fitted. Volume 13 , Issue 3.


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