As I mentioned in the Introduction, the world would be a very different place without curves. However, computers are very odd places, and not simply because that computers have no curves either. It's as weird as Steve Savitsky's song, "The World Inside the Crystal", depicts it.
The reason that there are no curves in computers is that the world of the computer is digital. It is made up of counted things, and counting is inherent inimical to curvature, which is the prototypical feature of the analog world, where a pencil can be tied to a string and a circle drawn easily. In a computer, a curve must needs be approximated; thankfully, the approximation can often be so close, so minutely carried out that the curve really looks like a curve to our eyes.
There are basicly two classes of curves in our universe - there are the classical conic curves - the circle, ellipse, parabola and hyperbola that are examined in analytical math; even straight lines can be considered to be members of this class. And then there are other curves. One class of these sorts of non-conic curves had long been designed with the help of a rubber or thin wood/metal ruler, called a spline, which could be bent at its ends, and which tended to distribute the curvature forced on it by the draftsman's fingers. In 1957-71, a pair of Frenchman, Paul de Faget de Casteljau at Citroen and Pierre Bezier at Renault were designing car bodies and other parts. The Frenchmen were interested in developing similar methods to create the curves on the new computers, and they succeeded. de Casteljau actually developed paper and pencil computational methods for spline curves first, but Citroen kept it secret for decades before Bezier repeated and extended his work 20 years later.
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| An infinity of ways to get from here to there. |
A spline, as used in Punch!, is a curve which passes through two points called handles and which is controlled by two other points, called the control points. Four points, then, define a simple curve. More complex curves can be assembled by stringing these splines end to end, so each pair shares one of the handles.
Each control point is associated with one of the handles of the Punch! figure. On an interior handle (one that is not at the end of an unclosed figure), there are actually two handles sharing the same point on the plane. The control point specifies two things about one end of the curve - the direction that the line takes immediately as it is drawn away from the handle, and also it specifies the "stiffness" of the curve from the handle to where it meets its partner from the other end. Since a point has freedom in two dimensions on a plane surface, the control points can control both quantities by their position relative to the handle they are associated with. Usually, the control point is plotted on a polar coordinate system relative to the handle. The polar angle defines the direction the line embarks on, and the distance from the handle defines the stiffness - 0 distance means the line has no stiffness, a longer line means higher stiffness.
Note a couple of things about this way of describing curvature. If the control point sits right on the handle, the line is not stiff, but like a weak rubber-band - the line emerging from the handle is straight and goes directly toward the next handle; therefore, straight lines are a class of Bezier curves, where the stiffness is zero and the direction doesn't matter. This is a very handy result. If there is distance between the handle and the control point, then the line emerges along the line from the handle towards the control point, but it starts veering off immediately - faster when the points are close together, more slowly when they are further apart. Lastly, if the control points on each side of a shared handle are at the same distances from the handle and on opposite sides, the curve through the handle has what a mathematician calls a continuous derivative, which is a rigorous definition of smoothness. It is often valuable that the control points be associated, or linked, in this way to assure this smoothness.
One of the oddities of mathematics - the only place where conic curves and parametric curves (another name for the Bezier and like curve schemes) is that either can be used to describe straight lines!
Punch! in some instances allows a user to control a curve's stiffness. See, for example, the curved shapes drawn with curved tools on the Detail tab. Their properties at one of the curve's handles points can have its stiffness set as a value from 0 to 10. But Punch! doesn't allow the user to change the emergence angle of a point, and if that point must be moved, a kink often develops as the curve has to make a radical change in direction in a small space. This is what causes the path tool to seem so difficult to work with at times.
Punch!'s handles for an object define the handles for the spline (for shapes, these are shown in "object edit" mode rather than "points edit"). What Curved to Fit does is make the control points, as well as the handles, visible and editable, so they can be changed to modify the curve. It does this within the framework of the PlansPlus PowerTool. When PlansPlus is exited, it sends the changes to the shapes back to Punch!, and they are incorporated into the basic plans, so the changes become permanent.
Curved to Fit and the contents of this help file are
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