The math behind a “drag to scale this line” operation.

LineScaling.swift

extension Line {
	
	/** Returns a scaled line according to the projection of `scalingPoint` onto the receiver.
		
	- parameter scalingPoint: A point interpreted as a vector used for scaling this line. This point should be something like the location of a touch or cursor during a "drag to scale" operation, but the point doesn't have to lie along the line itself. The point **must** be in the same coordinate space as the line.
	*/
	func lineByScalingToTouchPoint(scalingPoint: Point) -> Line {
		
		// This algorithm works by 
		// - treating the incoming point as a vector,
		// - treating the line (self) as a vector along an infinite line
		// - offsetting both vectors by the start point of the line
		//		- so the line has 0 as its origin to make math easier
		// - projecting the point vector onto the line
		// - finally offseting the projected vector BACK by the start amount of the line.
		// Projecting onto self is like casting a shadow of the scaling point onto our line.
		// That shadow is our scaled line.
		
		// Subtract the start from the end of the line and scaling point, so the line has a 0 origin.
		let lineVector = end - start
		let scalingPointOffsetByStart = scalingPoint - start
		
		// Projection(scalingVector) = constant * lineVector = ((scalingVector dot-product lineVector) / (lineVec dot-product itself)) * lineVec
		let vectorScalingConstantNumerator = scalingPointOffsetByStart.dotProduct(lineVector)
		let vectorScalingConstantDenomenator = lineVector.dotProduct(lineVector)
		let vectorScalingConstant = vectorScalingConstantNumerator / vectorScalingConstantDenomenator
		
		// see https://www.youtube.com/watch?v=27vT-NWuw0M
		let projectionVector = vectorScalingConstant * lineVector
		
		return Line(start: start, end: projectionVector + start)
	}
}