Normal Line Calculator

The normal line at a point on a curve is the line perpendicular to the tangent at that point. If the tangent has slope m_t, the normal has slope m_n = -1/m_t. That reciprocal-and-flip relationship is the defining property of perpendicular lines.

Once you know the slope of the normal, finding the equation is straightforward. You have a point (x0, f(x0)) on the curve and a slope m_n, so the line is y - f(x0) = m_n*(x - x0), which rearranges to y = m_n*x + b.

The special cases matter. If the tangent is horizontal (slope zero), the normal is vertical, and the normal line equation becomes x = x0 with no y-intercept to speak of. Conversely, if the tangent is vertical, the normal is horizontal with zero slope.

This calculator handles five function types. For a polynomial ax^m, the derivative at x0 is max0^(m-1). For sin(bx), it is bcos(bx0). For cos(bx), it is -bsin(bx0). For e^x, the derivative equals f(x0) itself. For ln(x), it is 1/x0.

Normal lines come up in optics (light reflecting off a curved mirror is governed by the angle the ray makes with the normal, not the tangent), in curve sketching, and in optimization problems where you need to project a point onto a curve by finding where the normal from the point intersects the curve.

One practical note: the tangent and normal are both local concepts. They only describe the curve at the single point x0. A short distance away, the curve may behave very differently from either line.

If the function is not differentiable at x0, such as a cusp or a corner, neither the tangent nor the normal is defined in the classical sense. The functions here are all smooth, so that issue does not arise.