We suppose that the curve is smooth, so that has tangents at each point. The tangent is in the direction of the vector at time ; in particular, a line that is tangent to at has gradient
If the moving point were to fly off , then it would leave in the direction . The vector is the velocity vector, which has components
speed is the magnitude of velocity.
The normal to at is the line through that is perpendicular to the tangent, hence has gradient .