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In analytic geometry, the **Hesse normal form** (named after Otto Hesse) is an equation used to describe a line in the Euclidean plane , a plane in Euclidean space , or a hyperplane in higher dimensions.^{[1]}^{[2]} It is primarily used for calculating distances (see point-plane distance and point-line distance).

It is written in vector notation as

The dot indicates the dot product (or scalar product).
Vector points from the origin of the coordinate system, *O*, to any point *P* that lies precisely in plane or on line *E*. The vector represents the unit normal vector of plane or line *E*. The distance is the shortest distance from the origin *O* to the plane or line.

Note: For simplicity, the following derivation discusses the 3D case. However, it is also applicable in 2D.

In the normal form,

a plane is given by a normal vector as well as an arbitrary position vector of a point . The direction of is chosen to satisfy the following inequality

By dividing the normal vector by its magnitude , we obtain the unit (or normalized) normal vector

and the above equation can be rewritten as

Substituting

we obtain the Hesse normal form

In this diagram, *d* is the distance from the origin. Because holds for every point in the plane, it is also true at point *Q* (the point where the vector from the origin meets the plane E), with , per the definition of the Scalar product

The magnitude of is the shortest distance from the origin to the plane.

The Quadrance (distance squared) from a line to a point is

If has unit length then this becomes

**^**Bôcher, Maxime (1915),*Plane Analytic Geometry: With Introductory Chapters on the Differential Calculus*, H. Holt, p. 44.**^**John Vince:*Geometry for Computer Graphics*. Springer, 2005, ISBN 9781852338343, pp. 42, 58, 135, 273

- Weisstein, Eric W. "Hessian Normal Form".
*MathWorld*.