track 2D offset after rotation

Question

red is my origin point, blue is my offset, both are 2D vectors, how can calculate blue after rotation?

this is the code I've been trying(just throwing things against a wall to see what sticks)

public static Vector2F CalcPointWithRotation(Vector2F origin, Vector2F point, float rotation)
{
    return new Vector2F((float)Math.Cos(rotation) * origin.x, (float)Math.Sin(rotation) * origin.y);
}

Answer 1

Assuming your rotation variable is an angle in radians around your object's origin point, and your point variable is an offset vector relative to the origin in the object's local coordinate space, then this is simply:

public static Vector2F CalcPointWithRotation(Vector2F origin, Vector2F point, float rotation)
{
    float c = (float)Math.Cos(rotation);
    float s = (float)Math.Sin(rotation);

    return new Vector2F(
                        origin.x + c * point.x - s * point.y,
                        origin.y + s * point.x + c * point.y,
    );
}

You may recognize this as a matrix multiplication:

$$\vec p ^\prime = \begin{bmatrix} \cos\theta &-\sin\theta\\\sin \theta & cos\theta \end{bmatrix}\vec p + \vec o$$

Or with a homogeneous matrix...

$$\begin{bmatrix}p^\prime_x \\ p^\prime_y \\ 1 \end{bmatrix} = \begin{bmatrix} \cos\theta &-\sin\theta & o_x\\\sin \theta & cos\theta & o_y \\ 0 & 0 & 1\end{bmatrix} \begin{bmatrix}p_x \\ p_y \\ 1 \end{bmatrix}$$

When rotation is 0 (or 2 pi, 4 pi, etc.) then this just gives origin + point. As the rotation value increases, the offset from the origin swings around, tracing a circle around that point.