Related Info: Quickdraw

Bezier Curve Ahead!

By David W. Smith, Los Gatos, CA

There comes a time in the development of some applications when arcs and wedges just don’t cut the mustard. You want to draw a pretty curve from point A to point B, and QuickDraw isn’t giving you any help. It seems like a good time to reach for a computer graphics text, blow the dust off of your college math, and try to decipher their explanation of splines. Stop. All is not lost. The Bezier curve may be just what you need.

Bezier Curves

Bezier curves (pronounced “bez-yeah”, after their inventor, a French mathematician) are well suited to graphics applications on the Macintosh for a number of reasons. First, they’re simple to describe. A curve is a function of four points. Second, the curve is efficient to calculate. From a precomputed table, the segments of the curve can be produced using only fixed-point multiplication. No trig, no messy quadratics, and no inSANEity. Third, and, to some, the most important, the Bezier curve is directly supported by the PostScript curve and curveto operators, and is one of the components of PostScript’s outlined fonts. The Bezier curve is also one of the principle drawing elements of Adobe Illustrator™. (Recently, they’ve shown up in a number of other places.)

Bezier curves have some interesting properties. Unlike some other classes of curves, they can fold over on themselves. They can also be joined together to form smooth (continuous) shapes. Figure 1 shows a few Bezier curves, including two that are joined to form a smooth shape.

The Gruesome Details

The description of Bezier curves below is going to get a bit technical. If you’re not comfortable with the math, you can trust that the algorithm works, and skip ahead to the implementation. However, if you’re curious about how the curves work and how to optimize their implementation, or just don’t trust using code that you don’t understand, read on.

The Bezier curve is a parametric function of four points; two endpoints and two “control” points. The curve connects the endpoints, but doesn’t necessarily touch the control points. The general form Bezier equation, which describes each point on the curve as a function of time, is:

where P1 and P4 are the endpoints, P2 and P3 are the control points, and the wn’s are weighting functions, which blend the four points to produce the curve. (The weights are applied to the h and v components of each point independently.) The single parameter t represents time, and varies from 0 to 1. The full form of the Bezier curve is:

We know that the curve touches each endpoint, so it isn’t too surprising that at t=0 the first weighting function is 1 and all others are 0 (i.e., the initial point on the curve is the first endpoint). Likewise, at t=1, the fourth weighting function is 1 and the rest are 0. However, it’s what happens between 0 and 1 that’s really interesting. A quick side-trip into calculus to take some first derivatives tells us that the second weighting function is maximized (has its greatest impact on the curve) at t=1/3, and the third weight is maximized at t=2/3. But the clever part--the bit that the graphics books don’t bother to mention--run the curve backwards by solving the equation for 1-t, and you find that w1(t)=w4(1-t) and w2(t)=w3(1-t). As we’ll see below, this symmetry halves the effort needed to compute values for the weights.

Figure 1. Some Beizer Curves and Shapes

Implementing Bezier Curves

One strategy for implementing Bezier curves is to divide the curve into a fixed number of segments and then to pre-compute the values of the weighting functions for each of the segments. The greater the number of segments, the smoother the curve. (I’ve found that 16 works well for display purposes, but 32 is better for hardcopy.) Computing any given curve becomes a simple matter of using the four points and the precomputed weights to produce the end-points of the curve segments. Fixed-point math yields reasonable accuracy, and is a hands down winner over SANE on the older (pre-Mac II) Macs, so we’ll use it.

We can optimize the process a bit. The curve touches each endpoint, so we can assume weights of 0 or 1 and needn’t compute weights for these points. Another optimization saves both time and space. By taking advantage of the symmetric nature of the Bezier equation, we can compute arrays of values for the first two of the weighting functions, and obtain values for the other two weights by indexing backwards into the arrays.

Drawing the curve, given the endpoints of the segments, is the duty of QuickDraw (or of PostScript, if you’re really hacking).

The listing below shows a reasonably efficient implementation of Bezier curves in Lightspeed C™. A few reminders about fixed-point math: an integer times a fixed-point number yields a fixed-point number, and a fixed by fixed multiplication uses a trap. The storage requirement for the algorithm, assuming 16 segments, (32 fixed-point values), is around 32*4*4, or 512 bytes. The algorithm computes all of the segments before drawing them so that the drawing can be done at full speed. (Having all of the segments around at one time can be useful for other reasons.)

More Fun With Curves

Given an implementation for Bezier curves, there are some neat things that fall out for almost free. Drawing a set of joined curves within an OpenPoly/ClosePoly or an OpenRgn/CloseRgn envelope yields an object that can be filled with a pattern. (Shades of popular illustration packages?) For that matter, lines, arcs, wedges, and Bezier curves can be joined to produce complicated shapes, such as outlined fonts. Given the direct mapping to PostScript’s curve and curveto operators, Bezier curves are a natural for taking better advantage of the LaserWriter.

As mentioned above, Bezier curves can be joined smoothly to produce more complicated shapes (see figure 1). The catch is that the point at which two curves are joined, and the adjacent control points, must be colinear (i.e., the three points must lay on a line). If you take a close look at Adobe Illustrator’s drawing tool, you’ll see what this means.

One nonobvious use of Bezier curves is in animation. The endpoints of the segments can be used as anchor-points for redrawing an object, giving it the effect of moving smoothly along the curve. One backgammon program that I’ve seen moves the tiles along invisible Bezier curves, and the effect is very impressive. For animation, you would probably want to vary the number of segments. Fortunately, the algorithm below is easily rewritten to produce the nth segment of an m segment curve given the the end and control points.

Further Optimizations

If you’re really tight on space or pressed for speed, there are a few things that you can do to tighten up the algorithm. A bit of code space (and a negligible amount of time) can be preserved by eliminating the setup code in favor of statically initializing the weight arrays with precomputed constant values. Drawing can be optimized by using GetTrapAddress to find the address in ROM of lineto, and then by calling it directly from inline assembly language, bypassing the trap mechanism. I’ve found that neither optimization is necessary for reasonable performance.

/*
**  Bezier  --  Support for Bezier curves
** Herein reside support routines for drawing Bezier curves.
**  Copyright (C) 1987, 1988 David W. Smith
**  Submitted to MacTutor for their source-disk.
*/

#include <MacTypes.h>
/*
   The greater the number of curve segments, the smoother the curve, 
and the longer it takes to generate and draw.  The number below was pulled 
out of a hat, and seems to work o.k.
 */
#define SEGMENTS 16

static Fixedweight1[SEGMENTS + 1];
static Fixedweight2[SEGMENTS + 1];

#define w1(s)  weight1[s]
#define w2(s)  weight2[s]
#define w3(s)  weight2[SEGMENTS - s]
#define w4(s)  weight1[SEGMENTS - s]

/*
 *  SetupBezier  --  one-time setup code.
 * Compute the weights for the Bezier function.
 *  For the those concerned with space, the tables can be precomputed. 
Setup is done here for purposes of illustration.
 */
void
SetupBezier()
{
 Fixed  t, zero, one;
 int    s;

 zero  = FixRatio(0, 1);
 one   = FixRatio(1, 1);
 weight1[0] = one;
 weight2[0] = zero;
 for ( s = 1 ; s < SEGMENTS ; ++s ) {
 t = FixRatio(s, SEGMENTS);
 weight1[s] = FixMul(one - t, FixMul(one - t, one - t));
 weight2[s] = 3 * FixMul(t, FixMul(t - one, t - one));
 }
 weight1[SEGMENTS] = zero;
 weight2[SEGMENTS] = zero;
}

/*
 *  computeSegments  --  compute segments for the Bezier curve
 * Compute the segments along the curve.
 *  The curve touches the endpoints, so don’t bother to compute them.
 */
static void
computeSegments(p1, p2, p3, p4, segment)
 Point  p1, p2, p3, p4;
 Point  segment[];
{
 int    s;
 
 segment[0] = p1;
 for ( s = 1 ; s < SEGMENTS ; ++s ) {
 segment[s].v = FixRound(w1(s) * p1.v + w2(s) * p2.v +
 w3(s) * p3.v + w4(s) * p4.v);
 segment[s].h = FixRound(w1(s) * p1.h + w2(s) * p2.h +
 w3(s) * p3.h + w4(s) * p4.h);
 }
 segment[SEGMENTS] = p4;
}

/*
 *  BezierCurve  --  Draw a Bezier Curve
 * Draw a curve with the given endpoints (p1, p4), and the given 
 * control points (p2, p3).
 *  Note that we make no assumptions about pen or pen mode.
 */
void
BezierCurve(p1, p2, p3, p4)
 Point  p1, p2, p3, p4;
{
 int    s;
 Point  segment[SEGMENTS + 1];

 computeSegments(p1, p2, p3, p4, segment);
 MoveTo(segment[0].h, segment[0].v);

 for ( s = 1 ; s <= SEGMENTS ; ++s ) {
 if ( segment[s].h != segment[s - 1].h ||
  segment[s].v != segment[s - 1].v ) {
 LineTo(segment[s].h, segment[s].v);
 }
 }
}

/*
**  CurveLayer.c  
** These routines provide a layer of support between my bare-  
 bones application skeleton and the Bezier curve code.   
  There’s little here of interest outside of the mouse 
  tracking and the curve drawing.
**  David W. Smith
*/

#include “QuickDraw.h”
#include “MacTypes.h”
#include “FontMgr.h”
#include “WindowMgr.h”
#include “MenuMgr.h”
#include “TextEdit.h”
#include “DialogMgr.h”
#include “EventMgr.h”
#include “DeskMgr.h”
#include “FileMgr.h”
#include “ToolboxUtil.h”
#include “ControlMgr.h”

/*
 *  Tracker objects.  Similar to MacAPP trackers, but much,
 much simpler.
 */
struct Tracker
{
 void (*track)();
 int    thePoint;
};

static struct Tracker aTracker;
static struct Tracker bTracker;

/*
 *  The Bezier curve control points.
 */
Point   control[4] = {{144,72}, {72,144}, {216,144}, {144,216}};


/*
 *  Draw
 *  Called from the skeleton to update the window.  Draw the   
 initial curve.
 */
Draw()
{
 PenMode(patXor);
 DrawTheCurve(control, true);
}

/*
 *  DrawTheCurve
 * Draw the given Bezier curve in the current pen mode.Draw 
   the control points if requested.
 */
DrawTheCurve(c, drawPoints)
 Point  c[];
{
 if ( drawPoints )
 DrawThePoints(c);
 BezierCurve(c[0], c[1], c[2], c[3]);
}

/*
 *  DrawThePoints
 *  Draw all of the control points.
 */
DrawThePoints(c)
 Point  c[];
{
 int    n;
 
 for ( n = 0 ; n < 4 ; ++n ) {
 DrawPoint(c, n);
 }
}

/*
 *  DrawPoint
 *  Draw a single control point
 */
DrawPoint(c, n)
 Point  c[];
 int    n;
{
 PenSize(3, 3);
 MoveTo(c[n].h - 1, c[n].v - 1);
 LineTo(c[n].h - 1, c[n].v - 1);
 PenSize(1, 1);
}

/*
 * GetTracker
 * Produce a tracker object
 * Called by the skeleton to handle mouse-down events.
 * If the mouse touches a control point, return a tracker for
 that point. Otherwise, return a tracker that drags a gray 
 rectangle.
 */
struct Tracker *
GetTracker(point)
 Point  point;
{
 void   TrackPoint(), TrackSelect();
 int    i;

 aTracker.track = TrackPoint;

 for ( i = 0 ; i < 4 ; ++i ) {
 if ( TouchPoint(control[i], point) ) {
 aTracker.thePoint = i;
 return (&aTracker);
 }
 }
 bTracker.track = TrackSelect;
 return (&bTracker);
}

/*
 *  TouchPoint
 *  Do the points touch?
 */
#define abs(a) (a < 0 ? -(a) : (a))

TouchPoint(target, point)
 Point  target;
 Point  point;
{
 SubPt(point, &target);
 if ( abs(target.h) < 3 && abs(target.v) < 3 )
 return (1);
 return (0);
}

/*
 *  TrackPoint
 *  Called while dragging a control point.
 */
void
TrackPoint(tracker, point, phase)
 struct Tracker  *tracker;
 Point  point;
 int    phase;
{
 Point  savePoint;

 switch ( phase ) {
 case 1:
 /* initial click - XOR out the control point */
 DrawPoint(control, tracker->thePoint);
 break;
 case 2:
 /* drag - undraw the original curve and draw the new one */
 DrawTheCurve(control, false);
 control[tracker->thePoint] = point;
 DrawTheCurve(control, false);
 break;
 case 3:
 /* release - redraw the control point */
 DrawPoint(control, tracker->thePoint);
 break;
 }
}

/*
 *  TrackSelect
 *  Track a gray selection rectangle
 */
static Pointfirst;
static Rect r;

void
TrackSelect(tracker, point, phase)
 struct Tracker  *tracker;
 Point  point;
 int    phase;
{
 switch ( phase ) {
 case 1:
 PenPat(gray);
 first = point;
 SetupRect(&r, first, point);
 FrameRect(&r);
 break;
 case 2:
 FrameRect(&r);
 SetupRect(&r, first, point);
 FrameRect(&r);
 break;
 case 3:
 FrameRect(&r);
 PenPat(black);
 break;
 }
}

/*
 *  SetupRect
 *  Setup the rectangle for tracking.
 */
#define min(x, y) (((x) < (y)) ? (x) : (y))
#define max(x, y) (((x) > (y)) ? (x) : (y))

SetupRect(rect, point1, point2)
 Rect   *rect;
 Point  point1;
 Point  point2;
{
 SetRect(rect,
 min(point1.h, point2.h),
 min(point1.v, point2.v),
 max(point1.h, point2.h),
 max(point1.v, point2.v));
}

/*
**  Skeleton.c  --  A bare-bones skeleton.
** This has been hacked up to demonstrate Bezier curves.  
    Other than the tracking technique, there’s little here of 
    interest.
**  David W. Smith
*/

#include “QuickDraw.h”
#include “MacTypes.h”
#include “FontMgr.h”
#include “WindowMgr.h”
#include “MenuMgr.h”
#include “TextEdit.h”
#include “DialogMgr.h”
#include “EventMgr.h”
#include “DeskMgr.h”
#include “FileMgr.h”
#include “ToolboxUtil.h”
#include “ControlMgr.h”

WindowRecordwRecord;
WindowPtr myWindow;

/*
 *  main
 *  Initialize the world, then handle events until told to quit.
 */
main() 
{
 InitGraf(&thePort);
 InitFonts();
 FlushEvents(everyEvent, 0);
 InitWindows();
 InitMenus();
 InitDialogs(0L);
 InitCursor();
 MaxApplZone();

 SetupMenus();
 SetupWindow();
 SetupBezier();

 while ( DoEvent(everyEvent) )
 ;
}

/*
 *  SetupMenus
 *  For the purpose of this demo, we get somewhat non-standard and use 
no menus.  Closing the window quits.
 */
SetupMenus()
{
 DrawMenuBar();
}

/*
 *  SetupWindow
 *  Setup the window for the Bezier demo.
 */
SetupWindow()
{
 Rect   bounds;

 bounds = WMgrPort->portBits.bounds;
 bounds.top += 36;
 InsetRect(&bounds, 5, 5);

 myWindow = NewWindow(&wRecord, &bounds, “\pBezier Sampler - Click and 
Drag”, 1, noGrowDocProc, 0L, 1, 0L);
 
 SetPort(myWindow);
}

/*
 *  DoEvent
 *  Generic event handling.
 */
DoEvent(eventMask)
 int    eventMask;
{
 EventRecordmyEvent;
 WindowPtrwhichWindow;
 Rect   r;
 
 SystemTask();
 if ( GetNextEvent(eventMask, &myEvent) )
 {
 switch ( myEvent.what )
 {
 case mouseDown:
 switch ( FindWindow( myEvent.where, &whichWindow ) )
 {
 case inDesk: 
 break;
 case inGoAway:
 if ( TrackGoAway(myWindow, myEvent.where) )
 {
 HideWindow(myWindow);
 return (0);
 }
 break;
 case inMenuBar:
 return (DoCommand(MenuSelect(myEvent.where)));
 case inSysWindow:
 SystemClick(&myEvent, whichWindow);
 break;
 case inDrag:
 break;
 case inGrow:
 break;
 case inContent:
 DoContent(&myEvent);
 break;
 default:
 break;;
 }
 break;
 case keyDown:
 case autoKey: 
 break;
 case activateEvt:
 break;
 case updateEvt:
 DoUpdate();
 break;
 default:
 break;
 }
 }
 return(1);
}

/*
 *  DoCommand
 *  Command handling would normally go here.
 */
DoCommand(mResult)
 long   mResult;
{
 int    theItem, temp;
 Str255 name;
 WindowPeek wPtr;
 
 theItem = LoWord(mResult);

 switch ( HiWord(mResult) )
 {
 }

 HiliteMenu(0);
 return(1);
}

/*
 *  DoUpdate
 *  Generic update handler.
 */
DoUpdate()
{
 BeginUpdate(myWindow);
 Draw();
 EndUpdate(myWindow);
}

/*
 *  DoContent
 *  Handle mouse-downs in the content area by asking the application 
to produce a tracker object.  We then call the tracker repeatedly to 
track the mouse. This technique came originally (as nearly as I can tell) 
from Xerox, and is used in a modified form in MacApp.
 */
struct Tracker
{
 int    (*Track)();
};

int
DoContent(pEvent)
 EventRecord*pEvent;
{
 struct Tracker  *GetTracker();
 struct Tracker  *t;
 Point  point, newPoint;
 
 point = pEvent->where;
 GlobalToLocal(&point);
 t = GetTracker(point);
 if ( t ) {
 (*t->Track)(t, point, 1);
 while ( StillDown() ) {
 GetMouse(&newPoint);
 if ( newPoint.h != point.h || newPoint.v != point.v ) {
 point = newPoint;
 (*t->Track)(t, point, 2);
 }
 }
 (*t->Track)(t, point, 3);
 }
}