重复旋转点绕枢轴点

Question

For a while now I've been using the following function to rotate a series of Points around a pivot point in various programs of mine.

private Point RotatePoint(Point point, Point pivot, double radians)
{
    var cosTheta = Math.Cos(radians);
    var sinTheta = Math.Sin(radians);

    var x = (cosTheta * (point.X - pivot.X) - sinTheta * (point.Y - pivot.Y) + pivot.X);
    var y = (sinTheta * (point.X - pivot.X) + cosTheta * (point.Y - pivot.Y) + pivot.Y);

    return new Point((int)x, (int)y);
}

This has always worked great, until I tried to rotate a shape repeatedly by small amounts. For example, this is what I get from calling it for 45° on a rectangular polygon made up of 4 points:

foreach (var point in points)
    Rotate(point, center, Math.PI / 180f * 45);

But this is what I get by calling rotate 45 times for 1°:

for (var i = 0; i < 45; ++i)
    foreach (var point in points)
        Rotate(point, center, Math.PI / 180f * 1)

As long as I call it only once it's fine, and it also seems like it gets gradually worse the lower the rotation degree is. Is there some flaw in the function, or am I misunderstanding something fundamental about what this function does?

How could I rotate repeatedly by small amounts? I could save the base points and use them to update the current points whenever the rotation changes, but is that the only way?

Answer 1

As TaW correctly pointed out, your Point position measure is off because of the integer rounding generated by the RotatePoint() method.

A simple correction in the method returned value, using float coordinates, will produce the correct measure:

To test it, create a Timer and register its Tick event as RotateTimerTick() :

PointF PivotPoint = new PointF(100F, 100F);
PointF RotatingPoint = new PointF(50F, 100F);
double RotationSpin = 0D;

private PointF RotatePoint(PointF point, PointF pivot, double radians)
{
    var cosTheta = Math.Cos(radians);
    var sinTheta = Math.Sin(radians);

    var x = (cosTheta * (point.X - pivot.X) - sinTheta * (point.Y - pivot.Y) + pivot.X);
    var y = (sinTheta * (point.X - pivot.X) + cosTheta * (point.Y - pivot.Y) + pivot.Y);

    return new PointF((float)x, (float)y);
}

private void RotateTimerTick(object sender, EventArgs e)
{
    RotationSpin += .5;

    if (RotationSpin > 90) RotationSpin = 0;
    RotatingPoint = RotatePoint(RotatingPoint, PivotPoint, (Math.PI / 180f) * RotationSpin);
    Panel1.Invalidate(new Rectangle(new Point(50,50), new Size(110, 110)));
}

private void Panel1_Paint(object sender, PaintEventArgs e)
{
    e.Graphics.SmoothingMode = SmoothingMode.AntiAlias;
    e.Graphics.FillEllipse(Brushes.White, new RectangleF(100, 100, 8, 8));
    e.Graphics.FillEllipse(Brushes.Yellow, new RectangleF(RotatingPoint, new SizeF(8, 8)));
}

This is the result using float values:

And this is what happens using integer values:

Answer 2

If you want you can use the Media3D to only deal with matrix and simplify the coding. Something as simple as this will work.

public Point3D Rotate(Point3D point, Point3D rotationCenter, Vector3D rotation, double degree)
{
    // create empty matrix
    var matrix = new Matrix3D();

    // translate matrix to rotation point
    matrix.Translate(rotationCenter - new Point3D());

    // rotate it the way we need
    matrix.Rotate(new Quaternion(rotation, degree));

    // apply the matrix to our point
    point = matrix.Transform(point);

    return point;
}

Then you simply call the method and specify the rotation. Lets say you work with 2D (like in your example) and lets assume we work with XY plane so the rotation is in Z. You can do something like :

var rotationPoint = new Point3D(0, 0, 0);
var currentPoint = new Point3D(10, 0, 0);

// rotate the current point around the rotation point in Z by 45 degree
var newPoint = Rotate(currentPoint, rotation, new Vector3D(0, 0, 1), 45d);