Orthogonal projection of a point onto a line

solve.py

def vec_scale(v: tuple[float, float], a: float):
    return (a * v[0], a * v[1])


def vec_mul_add(v1: tuple[float, float], v2: tuple[float, float], a: float = 1):
    """calc v1+a*v2"""
    return (v1[0] + a * v2[0], v1[1] + a * v2[1])


def inner_prod(v1: tuple[float, float], v2: tuple[float, float]):
    """v1^T * v2"""
    return v1[0] * v2[0] + v1[1] * v2[1]


def solve(coef: tuple[float, float, float], p: tuple[float, float]):
    """
    Params:
      coef: 3-tuple (a,b,c) to represent the linear equation ax+by+c=0.
      p: 2-tuple (px,py) to represent the point.
    """
    a, b, c = coef
    # find p0 at the line
    if b == 0:
        p0 = (-c / a, 0)
    else:
        p0 = (0, -c / b)

    # find v
    if a == 0:
        v = (1, 0)
    elif b == 0:
        v = (0, 1)
    else:
        v = (b, -a)
    return vec_mul_add(
        p0, vec_scale(v, inner_prod(vec_mul_add(p, p0, -1), v) / inner_prod(v, v))
    )


print(solve((-5, 3, 15), (-4, 5)))
# output: (3.3529411764705883, 0.5882352941176467)