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Hesse normal form

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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 $\mathbb {R} ^{2}$ , a plane in Euclidean space $\mathbb {R} ^{3}$ , 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

{\vec {r}}\cdot {\vec {n}}_{0}-d=0.\,

The dot $\cdot$ indicates the dot product (or scalar product). Vector ${\vec {r}}$ points from the origin of the coordinate system, O, to any point P that lies precisely in plane or on line E. The vector ${\vec {n}}_{0}$ represents the unit normal vector of plane or line E. The distance $d\geq 0$ is the shortest distance from the origin O to the plane or line.

Derivation/Calculation from the normal form

[edit]

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

In the normal form,

({\vec {r}}-{\vec {a}})\cdot {\vec {n}}=0\,

a plane is given by a normal vector ${\vec {n}}$ as well as an arbitrary position vector ${\vec {a}}$ of a point $A\in E$ . The direction of ${\vec {n}}$ is chosen to satisfy the following inequality

{\vec {a}}\cdot {\vec {n}}\geq 0\,

By dividing the normal vector ${\vec {n}}$ by its magnitude $|{\vec {n}}|$ , we obtain the unit (or normalized) normal vector

{\vec {n}}_{0}={{\vec {n}} \over {|{\vec {n}}|}}\,

and the above equation can be rewritten as

({\vec {r}}-{\vec {a}})\cdot {\vec {n}}_{0}=0.\,

Substituting

d={\vec {a}}\cdot {\vec {n}}_{0}\geq 0\,

we obtain the Hesse normal form

{\vec {r}}\cdot {\vec {n}}_{0}-d=0.\,

In this diagram, d is the distance from the origin. Because ${\vec {r}}\cdot {\vec {n}}_{0}=d$ 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 ${\vec {r}}={\vec {r}}_{s}$ , per the definition of the Scalar product

d={\vec {r}}_{s}\cdot {\vec {n}}_{0}=|{\vec {r}}_{s}|\cdot |{\vec {n}}_{0}|\cdot \cos(0^{\circ })=|{\vec {r}}_{s}|\cdot 1=|{\vec {r}}_{s}|.\,

The magnitude $|{\vec {r}}_{s}|$ of ${{\vec {r}}_{s}}$ is the shortest distance from the origin to the plane.

Distance to a line

[edit]

The Quadrance (distance squared) from a line $ax+by+c=0$ to a point $(x,y)$ is

{\frac {(ax+by+c)^{2}}{a^{2}+b^{2}}}.

If $(a,b)$ has unit length then this becomes $(ax+by+c)^{2}.$

References

[edit]

^ 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

External links

[edit]

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

Revisions and contributors Edit on Wikipedia Read on Wikipedia

View on Grokipedia

from Grokipedia

The Hesse normal form, also known as the Hessian normal form, is a standardized equation for representing a line in the plane, a plane in three-dimensional space, or more generally a hyperplane in Euclidean space of any dimension, using a unit normal vector and a constant term that denotes the signed perpendicular distance from the origin to the hyperplane.^[1]^[2] This form is derived from the general hyperplane equation

\mathbf{a} \cdot \mathbf{x} = b

by normalizing the normal vector

\mathbf{a}

to have unit length, yielding

\hat{\mathbf{a}} \cdot \mathbf{x} = d

, where

d = b / \|\mathbf{a}\|

and

\|\hat{\mathbf{a}}\| = 1

.^[1] The sign of

d

indicates on which side of the hyperplane the origin lies, with positive

d

meaning the origin is in the half-space pointed toward by

\hat{\mathbf{a}}

.^[2] Named after the 19th-century German mathematician Ludwig Otto Hesse, this representation simplifies geometric computations such as calculating distances from points to the hyperplane or determining parallelism between hyperplanes.^[2] For a plane in three dimensions, the equation takes the form

n_x x + n_y y + n_z z = p

, where

(n_x, n_y, n_z)

is the unit normal vector and

|p|

is the distance from the origin.^[1] The distance from a point

\mathbf{x}_0

to the plane is then given directly by

|\hat{\mathbf{n}} \cdot \mathbf{x}_0 - d|

, providing an efficient formula for applications in computer graphics, optimization, and computational geometry.^[1] In two dimensions, it parameterizes lines as

x \cos \varphi + y \sin \varphi = p

, with

\varphi

as the angle of the normal and

p

as the distance, offering a unique description up to the sign convention for

p > 0

.^[2]

Definition and notation

In two dimensions

In the Euclidean plane, the Hesse normal form represents a straight line using a unit normal vector and the perpendicular distance from the origin. The equation is given by

\cos \theta \, x + \sin \theta \, y - p = 0,

where

\theta

denotes the angle between the unit normal vector and the positive x-axis (measured counterclockwise), and

p \geq 0

is the perpendicular distance from the origin to the line.^[3] The normal vector

\vec{n} = (\cos \theta, \sin \theta)

=(cosθ,sinθ) has unit length $\|\vec{n}\| = 1$

∥=1.^[3] Geometrically, the line lies perpendicular to $\vec{n}$

and intersects the ray from the origin in the direction of $\vec{n}$

at a distance $p$ from the origin. The parameter $\theta$ specifies the orientation of this normal, which points away from the half-plane containing the origin (by the choice of normal direction to ensure $p \geq 0$ ). This parameterization uniquely identifies the line for $\theta \in [0, \pi)$ and $p \geq 0$ , avoiding redundancy in representation.^[3] For example, the line $x + y - 1 = 0$ has Hesse normal form $\frac{1}{\sqrt{2}} x + \frac{1}{\sqrt{2}} y - \frac{1}{\sqrt{2}} = 0$

1x+2

1y−2

1=0, where $\theta = 45^\circ$ (or $\pi/4$ radians) and $p = \frac{1}{\sqrt{2}}$

1 (approximately 0.707). Here, the unit normal $\vec{n} = \left( \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} \right)$

=(2

1,2

1) aligns with the line's original coefficients after normalization.^[4]

In three dimensions

In three dimensions, the Hesse normal form represents a plane in Euclidean 3-space

\mathbb{R}^3

using a unit normal vector

\vec{n}

and a scalar $p$ , given by the equation $\vec{n} \cdot \vec{x} = p$

⋅x

=p, where $\vec{x} = (x, y, z)^T$

=(x,y,z)T is any point on the plane.^[5] Here, $p$ denotes the signed distance from the origin to the plane along the direction of $\vec{n}$

.^[6] This vector notation extends the form from two dimensions, where lines are described similarly but in $\mathbb{R}^2$ .^[7] The normal vector $\vec{n} = (n_x, n_y, n_z)^T$

=(nx,ny,nz)T must satisfy the unit length condition $\|\vec{n}\| = 1$

∥=1, ensuring the coefficient's magnitude directly corresponds to the geometric distance.^[5] The components $n_x, n_y, n_z$ define the orientation of the plane perpendicular to $\vec{n}$

.^[6] An equivalent scalar form expands the dot product as $n_x x + n_y y + n_z z - p = 0$ .^[5] For example, consider the general plane equation $x + 2y + 2z - 3 = 0$ . To obtain the Hesse normal form, first compute the norm of the unnormalized normal vector $(1, 2, 2)$ , which is $\sqrt{1^2 + 2^2 + 2^2} = \sqrt{9} = 3$

=3. Dividing the entire equation by 3 yields $\frac{1}{3}x + \frac{2}{3}y + \frac{2}{3}z - 1 = 0$ , so $\vec{n} = \left(\frac{1}{3}, \frac{2}{3}, \frac{2}{3}\right)$

=(31,32,32) and $p = 1$ .

Derivation from the general equation

For lines in the plane

To derive the Hesse normal form of a line in the plane from its general equation

ax + by + c = 0

, where

a

and

b

are not both zero, begin by identifying the normal vector to the line as

(a, b)

. The magnitude of this vector provides the normalization factor

d = \sqrt{a^2 + b^2}

d=a2+b2

.^[8] Divide the entire general equation by $d$ to normalize the coefficients of $x$ and $y$ , resulting in the equation $\frac{a}{d} x + \frac{b}{d} y + \frac{c}{d} = 0.$ This form has a unit normal vector $\vec{n} = \left( \frac{a}{d}, \frac{b}{d} \right)$

=(da,db), since $\| \vec{n} \| = 1$

∥=1. The constant term $p = \frac{c}{d}$ represents the signed perpendicular distance from the origin to the line, allowing the equation to be expressed as $\vec{n} \cdot (x, y) + p = 0$

⋅(x,y)+p=0.^[8]^[5] The direction of $\vec{n}$

is ambiguous up to sign, as multiplying the original coefficients $a$ , $b$ , and $c$ by -1 reverses the normal and negates $p$ , while preserving the line. To resolve this, a convention is often adopted where $p \geq 0$ , achieved by flipping the sign of $\vec{n}$

(and thus $p$ ) if $p < 0$ ; the absolute value $|p|$ always gives the unsigned distance from the origin.^[8] For example, consider the general equation $3x - 4y + 5 = 0$ . Compute $d = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = 5$

=9+16

=5. Dividing through by 5 yields $\frac{3}{5} x - \frac{4}{5} y + 1 = 0,$ with $\vec{n} = \left( \frac{3}{5}, -\frac{4}{5} \right)$

=(53,−54) and $p = 1 > 0$ . No adjustment is needed, as $p \geq 0$ . This form maintains the Hesse normal equation $\vec{n} \cdot (x, y) + p = 0$

⋅(x,y)+p=0.^[8]

For planes in space

The general equation of a plane in three-dimensional Euclidean space is given by

ax + by + cz + d = 0

, where

(a, b, c)

are not all zero, defining the normal vector

\vec{N} = (a, b, c)

=(a,b,c) to the plane, and $d$ is a constant term.^[5]^[1] To obtain the Hesse normal form, the coefficients must be normalized such that the normal vector becomes a unit vector, facilitating interpretations like distance calculations. First, compute the Euclidean norm of the normal vector: $k = \sqrt{a^2 + b^2 + c^2}$

, which is nonzero since $\|\vec{N}\| \neq 0$

∥=0. Divide the entire equation by this norm to yield the normalized form: $\frac{a}{k} x + \frac{b}{k} y + \frac{c}{k} z + \frac{d}{k} = 0.$ This step ensures the coefficient vector $(a/k, b/k, c/k)$ has magnitude 1.^[5]^[1] Define the unit normal vector $\vec{n} = \frac{\vec{N}}{k} = \left( \frac{a}{k}, \frac{b}{k}, \frac{c}{k} \right)$

=kN

=(ka,kb,kc) and the constant $p = \frac{d}{k}$ . The equation can then be rewritten in vector notation as $\vec{n} \cdot \vec{x} + p = 0$

⋅x

+p=0, where $\vec{x} = (x, y, z)$

=(x,y,z) is the position vector of a point on the plane. Here, $p$ represents the signed perpendicular distance from the origin to the plane.^[5]^[9] The choice of direction for $\vec{n}$

is arbitrary up to sign, leading to two possible Hesse normal forms for the same plane: one with $\vec{n}$

and $p$ , and the other with $-\vec{n}$

and $-p$ . A common convention is to orient $\vec{n}$

such that $p \geq 0$ , which can be achieved by flipping the sign of $\vec{n}$

(and thus $p$ ) if necessary; this ensures the distance is nonnegative while preserving the plane's geometry.^[5]^[1] Consider the example of normalizing the plane equation $2x + y - 2z + 4 = 0$ . The normal vector is $\vec{N} = (2, 1, -2)$

=(2,1,−2), with norm $k = \sqrt{2^2 + 1^2 + (-2)^2} = \sqrt{9} = 3$

=3. Dividing by $k$ gives: $\frac{2}{3} x + \frac{1}{3} y - \frac{2}{3} z + \frac{4}{3} = 0,$ with $\vec{n} = \left( \frac{2}{3}, \frac{1}{3}, -\frac{2}{3} \right)$

=(32,31,−32) and $p = \frac{4}{3} > 0$ , so $\vec{n} \cdot \vec{x} + \frac{4}{3} = 0$

⋅x

+34=0. No flip is needed since $p \geq 0$ .^[5]

Geometric properties

Distance interpretation

In the Hesse normal form, the equation

\vec{n} \cdot \vec{x} = p

⋅x

=p of a line or plane, where $\vec{n}$

is a unit normal vector with $\|\vec{n}\| = 1$

∥=1, has the parameter $|p|$ representing the perpendicular distance from the origin to the line or plane.^[1]^[10] This interpretation applies uniformly in both two and three dimensions. In the plane, the distance from the origin to the line is $|p|$ . In three-dimensional space, the distance from the origin to the plane is likewise $|p|$ .^[2]^[11] To see this geometrically, consider the foot of the perpendicular from the origin to the hyperplane, which lies at the point $p \vec{n}$

. The Euclidean norm of this point is $\|p \vec{n}\| = |p| \|\vec{n}\| = |p|$

∥=∣p∣∥n

∥=∣p∣, confirming the distance.^[1]^[10] A special case occurs when $p = 0$ , in which the origin satisfies the equation and thus lies on the line or plane.^[1]

Signed distance and orientation

The parameter

p

in the Hesse normal form

\vec{n} \cdot \vec{x} = p

⋅x

=p, where $\vec{n}$

is the unit normal vector, represents the signed perpendicular distance from the origin to the line or plane. If $p > 0$ , the origin lies in the half-plane or half-space opposite to the direction pointed by $\vec{n}$

; conversely, if $p < 0$ , the origin is on the same side as $\vec{n}$

.^[4] The unit normal vector $\vec{n}$

establishes the orientation of the line or plane, directing towards the half-plane or half-space satisfying $\vec{n} \cdot \vec{x} > p$

⋅x

>p, which is conventionally regarded as the positive side. This orientation partitions the space into two regions relative to the line or plane.^[4]^[12] Each line or plane possesses two equivalent representations in Hesse normal form, arising from the arbitrary choice of $\vec{n}$

's direction: the form with $\vec{n}$

and $p$ , or the alternative with $-\vec{n}$

and $-p$ , which describe the identical geometric entity but with inverted orientation.^[4]^[2] In two dimensions, the orientation is parameterized by the angle $\theta$ of the normal vector, which is defined modulo $\pi$ rather than $2\pi$ , due to the equivalence of $\vec{n}$

and $-\vec{n}$

in representing the same line. For instance, a line given by $\vec{n} \cdot \vec{x} = p$

⋅x

=p transforms under flipping to $(-\vec{n}) \cdot \vec{x} = -p$

)⋅x

=−p, maintaining the line while reversing the positive half-plane.^[2]^[4]

Applications

Point-to-line distance

The perpendicular distance from a point to a line in the plane, when the line is expressed in Hesse normal form, leverages the form's inherent normalization to simplify computation. For a line given by

\cos \theta \, x + \sin \theta \, y = p

, where

(\cos \theta, \sin \theta)

is the unit normal vector and

p

is the signed distance from the origin to the line, the distance

d

from a point

(x_0, y_0)

to the line is

d = |\cos \theta \, x_0 + \sin \theta \, y_0 - p|

.^[1] This formula arises from the geometric interpretation of the Hesse normal form. The expression

\cos \theta \, x + \sin \theta \, y - p

represents the signed distance from the point

(x, y)

to the line along the direction of the unit normal vector; taking the absolute value yields the unsigned perpendicular distance.^[1] In contrast to the general line equation

ax + by + c = 0

, where the distance requires normalization by dividing by

\sqrt{a^2 + b^2}

a2+b2

, the Hesse form embeds this unit normalization, avoiding additional square root computations.^[2] For example, consider the line $\frac{1}{\sqrt{2}} x + \frac{1}{\sqrt{2}} y = 1$

1x+2

1y=1 and the point $(1, 1)$ . Substituting the point coordinates gives $\frac{1}{\sqrt{2}} \cdot 1 + \frac{1}{\sqrt{2}} \cdot 1 - 1 = \sqrt{2} - 1$

1⋅1+2

1⋅1−1=2

−1, so the distance is $|\sqrt{2} - 1| \approx 0.414$

−1∣≈0.414.

Point-to-plane distance

The Hesse normal form of a plane in three dimensions is given by

\vec{n} \cdot \vec{x} = p

⋅x

=p, where $\vec{n}$

is a unit normal vector to the plane and $p$ is a scalar representing the signed distance from the origin to the plane along the normal direction.^[5] The distance from a point $\vec{x_0}$

to this plane is the length of the perpendicular from $\vec{x_0}$

to the plane, which simplifies to $|\vec{n} \cdot \vec{x_0} - p|$

⋅x0

−p∣ due to the unit length of $\vec{n}$

.^[5] This formula arises because substituting $\vec{x_0}$

into the plane equation yields the signed projection of the vector from the origin to $\vec{x_0}$

onto $\vec{n}$

, adjusted by the plane's offset $p$ . The signed distance, $\vec{n} \cdot \vec{x_0} - p$

⋅x0

−p, preserves the direction: positive values indicate the point lies on the side of the plane toward which $\vec{n}$

points, while negative values indicate the opposite side.^[5] This orientation is useful for determining half-space membership in geometric computations. To derive this, consider a point $\vec{q}$

on the plane, so $\vec{n} \cdot \vec{q} = p$

⋅q

=p. The vector from $\vec{q}$

to $\vec{x_0}$

is $\overrightarrow{q x_0} = \vec{x_0} - \vec{q}$

=x0

−q

. The shortest distance is the projection of this vector onto $\vec{n}$

, given by $|\overrightarrow{q x_0} \cdot \vec{n}| / \|\vec{n}\|$

⋅n

∣/∥n

∥. Since $\|\vec{n}\| = 1$

∥=1 and $\overrightarrow{q x_0} \cdot \vec{n} = (\vec{x_0} - \vec{q}) \cdot \vec{n} = \vec{n} \cdot \vec{x_0} - p$

⋅n

=(x0

−q

)⋅n

⋅x0

−p, the distance simplifies to $|\vec{n} \cdot \vec{x_0} - p|$

⋅x0

−p∣. This projection follows from the geometric definition: the distance equals the component of $\overrightarrow{q x_0}$

along $\vec{n}$

, or $|\overrightarrow{q x_0}| \cos \theta$

∣cosθ, where $\theta$ is the angle between $\overrightarrow{q x_0}$

and $\vec{n}$

, and $\cos \theta = (\overrightarrow{q x_0} \cdot \vec{n}) / (|\overrightarrow{q x_0}| \|\vec{n}\|)$

⋅n

)/(∣qx0

∣∥n

∥).^[13] For example, consider the plane $\frac{1}{3} x + \frac{2}{3} y + \frac{2}{3} z = 1$ , where $\vec{n} = \left( \frac{1}{3}, \frac{2}{3}, \frac{2}{3} \right)$

=(31,32,32) (with $\|\vec{n}\| = 1$

∥=1) and $p = 1$ . The distance from the origin $(0,0,0)$ is $| \vec{n} \cdot (0,0,0) - 1 | = |0 - 1| = 1$

⋅(0,0,0)−1∣=∣0−1∣=1.

Uses in computational geometry

In computational geometry, the Hesse normal form simplifies the computation of intersections between lines or planes by normalizing the coefficients to unit length, which avoids scaling inconsistencies and facilitates direct solution of linear systems without additional normalization steps. For instance, when determining the intersection point of two lines in 2D, the normalized equations allow for straightforward Cramer's rule application or matrix inversion, reducing numerical instability in algorithms for ray tracing or polygon clipping. This property is particularly advantageous in geometric query processing, where multiple intersections must be resolved efficiently.^[14] The parameterization of lines in Hesse normal form using an angle

\theta

and signed distance

p

—where the line equation is

x \cos \theta + y \sin \theta = p

—is widely employed in robotics and computer graphics for robust line representation. This polar-like form avoids the singularities inherent in slope-intercept representations (e.g., vertical lines) and is integral to variants of the Hough transform for detecting lines in images, as introduced by Duda and Hart, enabling efficient accumulation in parameter space for edge detection tasks. In 3D extensions, the form parameterizes planes similarly, supporting plane detection in point clouds for applications like 3D reconstruction in graphics pipelines. For convex hull computation and collision detection, the Hesse normal form aids in representing half-planes as

\mathbf{n} \cdot \mathbf{x} \leq p

, where

\mathbf{n}

is a unit normal, allowing efficient intersection algorithms via linear programming solvers that leverage the normalized distances for bounding volume hierarchies and separating axis tests. This is evident in randomized incremental constructions for convex polyhedra, where half-plane intersections form the hull boundaries without coefficient rescaling. In higher dimensions, the generalization to hyperplanes—

\mathbf{n} \cdot \mathbf{x} = p

with

\|\mathbf{n}\| = 1

—proves useful in machine learning, particularly in support vector machines, where normalizing the weight vector to unit length interprets the bias as the hyperplane's offset from the origin, directly yielding geometric margins.

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