SummerCode - 1 year ago 99

iOS Question

I have only found this blog with a relevant answer http://seant23.wordpress.com/2010/11/12/offset-bezier-curves/ ,but unfortunately i don't know the language and can't understand the maths behind it. What i need is to know how to make a bezier curve parallel to the one that i have.

I have a Point, Segment and Path class, but i don't understand how to divide the path into segments.

The Point class has the CGPoint location public variable,

the Segment class has as properties 4 points, Point *control1, *control2, *point2 and*point1;

the Path class contains an NSMutableArray of segments and a Point startPoint.

I am new to objective c and i would really appreciate some help, if not for my specific class construction, at least for a more general method.

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Answer Source

I don't know about the specific problem you're solving, but one cute (and very easy) solution is to just render the outline outline of a bezier curve, e.g.:

This is easily done using Core Graphics (in this case, a `drawRect`

of a `UIView`

subclass):

```
- (void)drawRect:(CGRect)rect {
CGPathRef path = [self newBezierPath];
CGPathRef outlinePath = CGPathCreateCopyByStrokingPath(path, NULL, 10, kCGLineCapButt, kCGLineJoinBevel, 0);
CGContextRef context = UIGraphicsGetCurrentContext();
CGContextSetLineWidth(context, 3.0);
CGContextAddPath(context, outlinePath);
CGContextSetStrokeColorWithColor(context, [[UIColor redColor] CGColor]);
CGContextDrawPath(context, kCGPathStroke);
CGPathRelease(path);
CGPathRelease(outlinePath);
}
- (CGPathRef)newBezierPath {
CGPoint point1 = CGPointMake(10.0, 50.0);
CGPoint point2 = CGPointMake(self.bounds.size.width - 10.0, point1.y + 150.0);
CGPoint controlPoint1 = CGPointMake(point1.x + 400.0, point1.y);
CGPoint controlPoint2 = CGPointMake(point2.x - 400.0, point2.y);
CGMutablePathRef path = CGPathCreateMutable();
CGPathMoveToPoint(path, NULL, point1.x, point1.y);
CGPathAddCurveToPoint(path, NULL, controlPoint1.x, controlPoint1.y, controlPoint2.x, controlPoint2.y, point2.x, point2.y);
return path;
}
```

Or in Swift 3:

```
override func draw(_ rect: CGRect) {
let path = bezierPath().cgPath
let outlinePath = path.copy(strokingWithWidth: 10, lineCap: .butt, lineJoin: .bevel, miterLimit: 0)
let context = UIGraphicsGetCurrentContext()!
context.setLineWidth(3)
context.addPath(outlinePath)
context.setStrokeColor(UIColor.red.cgColor)
context.strokePath()
}
private func bezierPath() -> UIBezierPath {
let point1 = CGPoint(x: 10.0, y: 50.0)
let point2 = CGPoint(x: bounds.size.width - 10.0, y: point1.y + 150.0)
let controlPoint1 = CGPoint(x: point1.x + 400.0, y: point1.y)
let controlPoint2 = CGPoint(x: point2.x - 400.0, y: point2.y)
let path = UIBezierPath()
path.move(to: point1)
path.addCurve(to: point2, controlPoint1: controlPoint1, controlPoint2: controlPoint2)
return path
}
```

If you really want to draw a parallel path, that's more complicated. But you can render something like this (original bezier path in red, a "parallel" line in blue).

I'm not entirely sure about the algorithm you've identified, but I rendered this by

- calculating the bezier points (the red path) myself, breaking it up finely enough to make the effect smooth;
- calculate the angle between each point and the next;
- calculate the coordinates of the offset path (the blue one) by taking the points on the bezier path, and calculating new points that are perpendicular to the angle I just determined; and
- using those offset point coordinates, draw a new series of line segments to render the parallel line to the bezier.

Thus, in Objective-C, that might look like:

```
- (void)drawRect:(CGRect)rect {
CGPoint point1 = CGPointMake(10.0, 50.0);
CGPoint point2 = CGPointMake(self.bounds.size.width - 10.0, point1.y + 150.0);
CGPoint controlPoint1 = CGPointMake(point1.x + 400.0, point1.y);
CGPoint controlPoint2 = CGPointMake(point2.x - 400.0, point2.y);
// draw original bezier path in red
[[UIColor redColor] setStroke];
[[self bezierPathFromPoint1:point1
point2:point2
controlPoint1:controlPoint1
controlPoint2:controlPoint2] stroke];
// calculate and draw offset bezier curve in blue
[[UIColor blueColor] setStroke];
[[self offsetBezierPathBy:10.0
point1:point1
point2:point2
controlPoint1:controlPoint1
controlPoint2:controlPoint2] stroke];
}
- (UIBezierPath *)bezierPathFromPoint1:(CGPoint)point1
point2:(CGPoint)point2
controlPoint1:(CGPoint)controlPoint1
controlPoint2:(CGPoint)controlPoint2 {
UIBezierPath *path = [UIBezierPath bezierPath];
[path moveToPoint:point1];
[path addCurveToPoint:point2 controlPoint1:controlPoint1 controlPoint2:controlPoint2];
return path;
}
- (UIBezierPath *)offsetBezierPathBy:(CGFloat)offset
point1:(CGPoint)point1
point2:(CGPoint)point2
controlPoint1:(CGPoint)controlPoint1
controlPoint2:(CGPoint)controlPoint2 {
UIBezierPath *path = [UIBezierPath bezierPath];
static NSInteger numberOfPoints = 100;
CGPoint previousPoint = [self cubicBezierAtTime:0.0
point1:point1
point2:point2
controlPoint1:controlPoint1
controlPoint2:controlPoint2];
CGPoint point;
double angle;
for (NSInteger i = 1; i <= numberOfPoints; i++) {
double t = (double) i / numberOfPoints;
point = [self cubicBezierAtTime:t
point1:point1
point2:point2
controlPoint1:controlPoint1
controlPoint2:controlPoint2];
// calculate the angle to the offset point
// this is the angle between the two points, plus 90 degrees (pi / 2.0)
angle = atan2(point.y - previousPoint.y, point.x - previousPoint.x) + M_PI_2;
if (i == 1)
[path moveToPoint:[self offsetPoint:previousPoint by:offset angle:angle]];
previousPoint = point;
[path addLineToPoint:[self offsetPoint:previousPoint by:offset angle:angle]];
}
return path;
}
// return point offset by particular distance and particular angle
- (CGPoint)offsetPoint:(CGPoint)point by:(CGFloat)offset angle:(double)angle {
return CGPointMake(point.x + cos(angle) * offset, point.y + sin(angle) * offset);
}
// Manually calculate cubic bezier curve
//
// B(t) = (1-t)^3 * point1 + 3 * (1-t)^2 * t controlPoint1 + 3 * (1-t) * t^2 * pointPoint2 + t^3 * point2
- (CGPoint)cubicBezierAtTime:(double)t
point1:(CGPoint)point1
point2:(CGPoint)point2
controlPoint1:(CGPoint)controlPoint1
controlPoint2:(CGPoint)controlPoint2 {
double oneMinusT = 1.0 - t;
double oneMinusTSquared = oneMinusT * oneMinusT;
double oneMinusTCubed = oneMinusTSquared * oneMinusT;
double tSquared = t * t;
double tCubed = tSquared * t;
CGFloat x = point1.x * oneMinusTCubed +
3.0 * oneMinusTSquared * t * controlPoint1.x +
3.0 * oneMinusT * tSquared * controlPoint2.x +
tCubed * point2.x;
CGFloat y = point1.y * oneMinusTCubed +
3.0 * oneMinusTSquared * t * controlPoint1.y +
3.0 * oneMinusT * tSquared * controlPoint2.y +
tCubed * point2.y;
return CGPointMake(x, y);
}
```

Or, in Swift 3:

```
override func draw(_ rect: CGRect) {
let point1 = CGPoint(x: 10.0, y: 50.0)
let point2 = CGPoint(x: bounds.size.width - 10.0, y: point1.y + 150.0)
let controlPoint1 = CGPoint(x: point1.x + 400.0, y: point1.y)
let controlPoint2 = CGPoint(x: point2.x - 400.0, y: point2.y)
UIColor.red.setStroke()
bezierPath(from: point1, to: point2, withControl: controlPoint1, and: controlPoint2).stroke()
UIColor.blue.setStroke()
offSetBezierPath(by: 5, from: point1, to: point2, withControl: controlPoint1, and: controlPoint2).stroke()
}
private func bezierPath(from point1: CGPoint, to point2: CGPoint, withControl controlPoint1: CGPoint, and controlPoint2:CGPoint) -> UIBezierPath {
let path = UIBezierPath()
path.move(to: point1)
path.addCurve(to: point2, controlPoint1: controlPoint1, controlPoint2: controlPoint2)
return path
}
private func offSetBezierPath(by offset: CGFloat, from point1: CGPoint, to point2: CGPoint, withControl controlPoint1: CGPoint, and controlPoint2:CGPoint) -> UIBezierPath {
let path = UIBezierPath()
let numberOfPoints = 100
var previousPoint = cubicBezier(at: 0, point1: point1, point2: point2, controlPoint1: controlPoint1, controlPoint2: controlPoint2)
for i in 1 ... numberOfPoints {
let time = CGFloat(i) / CGFloat(numberOfPoints)
let point = cubicBezier(at: time, point1: point1, point2: point2, controlPoint1: controlPoint1, controlPoint2: controlPoint2)
// calculate the angle to the offset point
// this is the angle between the two points, plus 90 degrees (pi / 2.0)
let angle = atan2(point.y - previousPoint.y, point.x - previousPoint.x) + .pi / 2;
if i == 1 {
path.move(to: calculateOffset(of: previousPoint, by: offset, angle: angle))
}
previousPoint = point
path.addLine(to: calculateOffset(of: previousPoint, by: offset, angle: angle))
}
return path
}
/// Return point offset by particular distance and particular angle
///
/// - Parameters:
/// - point: Point to offset.
/// - offset: How much to offset by.
/// - angle: At what angle.
///
/// - Returns: New `CGPoint`.
private func calculateOffset(of point: CGPoint, by offset: CGFloat, angle: CGFloat) -> CGPoint {
return CGPoint(x: point.x + cos(angle) * offset, y: point.y + sin(angle) * offset)
}
/// Manually calculate cubic bezier curve
///
/// B(t) = (1-t)^3 * point1 + 3 * (1-t)^2 * t controlPoint1 + 3 * (1-t) * t^2 * pointPoint2 + t^3 * point2
///
/// - Parameters:
/// - time: Time, a value between zero and one.
/// - point1: Starting point.
/// - point2: Ending point.
/// - controlPoint1: First control point.
/// - controlPoint2: Second control point.
///
/// - Returns: Point on bezier curve.
private func cubicBezier(at time: CGFloat, point1: CGPoint, point2: CGPoint, controlPoint1: CGPoint, controlPoint2: CGPoint) -> CGPoint {
let oneMinusT = 1.0 - time
let oneMinusTSquared = oneMinusT * oneMinusT
let oneMinusTCubed = oneMinusTSquared * oneMinusT
let tSquared = time * time
let tCubed = tSquared * time
var x = point1.x * oneMinusTCubed
x += 3.0 * oneMinusTSquared * time * controlPoint1.x
x += 3.0 * oneMinusT * tSquared * controlPoint2.x
x += tCubed * point2.x
var y = point1.y * oneMinusTCubed
y += 3.0 * oneMinusTSquared * time * controlPoint1.y
y += 3.0 * oneMinusT * tSquared * controlPoint2.y
y += tCubed * point2.y
return CGPoint(x: x, y: y)
}
```

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