# Hyperplane
A hyperplane is a subspace, or a translated subspace, whose dimension is one less than that of the space containing it: in an \( n \)-dimensional vector space it is an \( (n-1) \)-dimensional subspace, and in an affine space it is an \( (n-1) \)-dimensional affine subspace[1](https://sanweb.lib.msu.edu/crcmath/math/math/h/h453.htm) • [2](https://tangente-mag.com/en/articles/hyperplane). In three dimensions a hyperplane is an ordinary plane, in two dimensions it is a line, and the term generalizes the plane to higher dimensions[3](https://mathworld.wolfram.com/Plane.html).
| Key fact | Detail |
|---|---|
| Dimension | In an \( n \)-dimensional space, a hyperplane has dimension \( n-1 \) and codimension 1[2](https://tangente-mag.com/en/articles/hyperplane) |
| Linear form | A hyperplane in a vector space \( E \) has the form \( \{x : f(x) = \lambda\} \) for a nonzero \( f \in E^* \) and \( \lambda \in K \)[4](https://encyclopediaofmath.org/wiki/Vector_space) |
| Kernel view | A linear hyperplane is the kernel of any nonzero linear map from the vector space to the underlying field; equivalently a subspace \( V \subset W \) with \( W/V \) one-dimensional[1](https://sanweb.lib.msu.edu/crcmath/math/math/h/h453.htm) |
| Equation | In \( \mathbb{R}^N \), the implicit equation is \( \mathbf{n} \cdot \mathbf{x} = \sum_{i=1}^{N} n_i x_i = d \), with \( \mathbf{n} \) the normal and \( d \) the signed distance from the origin in units of the normal's length[5](https://math.stackexchange.com/questions/4478169/is-a-hyper-plane-uniquely-defined-by-a-single-normal-vector) |
| Normal vector | The normal is unique up to nonzero scalar multiplication, because an \( (n-1) \)-dimensional hyperplane in \( \mathbb{R}^n \) leaves one dimension perpendicular to it[6](https://math.stackexchange.com/questions/2796428/does-every-hyperplane-have-a-unique-normal-vector-up-to-non-zero-scalar-multipl) |
| Affine case | A linear variety (a translate \( x_0 + L \) of a subspace \( L \)) of codimension 1 is called a hyperplane[7](https://encyclopediaofmath.org/wiki/Linear_variety) |
| Classification use | In a linear support vector machine the separating hyperplane is \( \mathbf{w}^T \mathbf{x} + b = 0 \), with margin \( \rho = 2/\|\mathbf{w}\|_2 \)[8](https://web.stanford.edu/class/cs276/handouts/lecture14-SVMs-handout-1-per.pdf) |
## Linear and affine hyperplanes
The linear case is the simplest. Let \( a_1, a_2, \ldots, a_n \) be scalars not all equal to 0. The set of vectors \( \mathbf{X} = (x_1, \ldots, x_n) \) in \( \mathbb{R}^n \) satisfying
\[ a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = 0 \]
is a subspace of \( \mathbb{R}^n \) called a hyperplane[1](https://sanweb.lib.msu.edu/crcmath/math/math/h/h453.htm). Because one independent equation removes exactly one degree of freedom, the solution set has dimension \( n-1 \)[2](https://tangente-mag.com/en/articles/hyperplane). Three equivalent characterizations follow from this: a hyperplane is any codimension-1 subspace, any subspace \( V \) of \( W \) for which the quotient \( W/V \) is one-dimensional, and the kernel of any nonzero linear map from the vector space to the underlying field[1](https://sanweb.lib.msu.edu/crcmath/math/math/h/h453.htm). In dual-space language, the kernel \( \{f\}_\perp \) of a nonzero \( f \in E^* \) is a maximal proper linear subspace, sometimes called a hypersubspace[4](https://encyclopediaofmath.org/wiki/Vector_space).
**Affine hyperplanes.** Most applications use the affine version, which need not pass through the origin. An affine hyperplane is a translate of a codimension-one linear subspace: choosing a base point \( p \) in the set \( H \), one writes \( H = p + L \), where \( L \) is the direction space[9](https://androma.org/theorems/13566). In the language of linear varieties, a subset \( M = x_0 + L \) of a vector space \( E \) is a linear variety, and the variety corresponding to a subspace of codimension 1 is called a hyperplane[7](https://encyclopediaofmath.org/wiki/Linear_variety). The nLab summarizes common usage: a hyperplane is an affine or linear subspace of an affine or linear space respectively, typically required to have positive dimension and codimension, and often required to have codimension exactly 1[10](https://ncatlab.org/nlab/show/hyperplane).
Affine functions give a coordinate-free description. If \( u \) is a non-constant affine function on an affine space \( E \) and \( A \) is a scalar, the set of \( x \) satisfying \( u(x) = A \) is a hyperplane; conversely, every hyperplane \( H \) in \( E \) satisfies \( H = u^{-1}(0) \) for some affine function \( u \), and every affine function with this property is a scalar multiple of a fixed one. Hyperplanes with equations \( u(x) = \alpha \) and \( u(x) = \beta \) for the same \( u \) are parallel[11](https://math.mit.edu/~hrm/palestine/bourbaki-algebra-1i.pdf).
## Equations and normal vectors
In Cartesian coordinates, a hyperplane in \( N \) dimensions is described by one linear equation,
\[ \mathbf{n} \cdot \mathbf{x} = \sum_{i=1}^{N} n_i x_i = d, \]
where \( \mathbf{n} = (n_1, \ldots, n_N) \) is the normal vector, \( \mathbf{x} \) is any point on the hyperplane, and \( d \) is the signed distance of the hyperplane from the origin in units of the normal's length[5](https://math.stackexchange.com/questions/4478169/is-a-hyper-plane-uniquely-defined-by-a-single-normal-vector). Equivalently, an affine function \( x \mapsto \beta_0 + \sum_{j=1}^{p} x_j \beta_j \) determines the hyperplane \( H = \{x : \sum_j x_j \beta_j + \beta_0 = 0\} \) together with the two half-spaces where the expression is positive or negative; the vector \( N = (\beta_1, \ldots, \beta_p) \) is the normal, and by scaling one can always choose \( N \) with \( \|N\|_2 = 1 \), scaling \( \beta_0 \) along with it to keep \( H \) unchanged[12](https://web.stanford.edu/class/stats202/notes/Support-vector-machines/Support-vector-machines.html?highlight=e+learning).
The signed quantity \( \beta_0 + x_1 \beta_1 + \cdots + x_p \beta_p \) measures how far a point sits from the hyperplane along the normal, and its sign tells which side of the hyperplane the point lies on[12](https://web.stanford.edu/class/stats202/notes/Support-vector-machines/Support-vector-machines.html?highlight=e+learning). The normal vector is unique up to multiplication by a nonzero scalar: a hyperplane is \( (n-1) \)-dimensional, which leaves only one dimension of directions perpendicular to it[6](https://math.stackexchange.com/questions/2796428/does-every-hyperplane-have-a-unique-normal-vector-up-to-non-zero-scalar-multipl). The equation degenerates if the normal is the zero vector: \( \mathbf{n} \cdot \mathbf{x} = d \) is then either nowhere true (when \( d \neq 0 \)) or everywhere true (when \( d = 0 \)), so it refers to nothing or to the entire space rather than to a hyperplane[5](https://math.stackexchange.com/questions/4478169/is-a-hyper-plane-uniquely-defined-by-a-single-normal-vector).
In the three-dimensional case of an ordinary plane with nonzero normal \( \mathbf{n} = (a, b, c) \) through the point \( \mathbf{x}_0 \), the equation \( \mathbf{n} \cdot (\mathbf{x} - \mathbf{x}_0) = 0 \) expands to \( ax + by + cz + d = 0 \) with \( d = -ax_0 - by_0 - cz_0 \), and the plane lies at distance \( D = d / \sqrt{a^2 + b^2 + c^2} \) from the origin[3](https://mathworld.wolfram.com/Plane.html).
## Hyperplane arrangements
A finite collection of codimension-one affine subspaces of an \( \ell \)-dimensional affine space over a field \( K \) is called an arrangement of hyperplanes[13](https://encyclopediaofmath.org/wiki/Arrangement_of_hyperplanes). Arrangements carry a combinatorial structure: an edge of the arrangement is a non-empty intersection of its hyperplanes, and the set of edges, partially ordered by reverse inclusion, forms a geometric semi-lattice whose rank is given by codimension[13](https://encyclopediaofmath.org/wiki/Arrangement_of_hyperplanes).
Several standard arrangements serve as reference examples. A subset of the coordinate hyperplanes is a Boolean arrangement; an arrangement is in general position if it is locally Boolean at each point; the braid arrangement consists of the hyperplanes \( \{x_i = x_j : 1 \le i < j \le \ell\} \), which are the reflecting hyperplanes of the symmetric group on \( \ell \) letters; and the reflecting hyperplanes of any finite reflection group form an arrangement[13](https://encyclopediaofmath.org/wiki/Arrangement_of_hyperplanes). The theory also links with De Concini-Procesi wonderful models, the Feichtner-Yuzvinsky algebra built from an atomic lattice, fatpoints, blowups of projective space, and singularities of plane curves[14](https://arxiv.org/abs/1101.0356). Computationally, arrangements are basic objects in computational geometry, and algorithms exist to enumerate their cells and vertices; one incremental cell-enumeration algorithm was found in computational experiments to be significantly faster than Avis and Fukuda's reverse search for most test cases[15](https://epubs.siam.org/doi/10.1137/15M1027930).
## Hyperplanes in classification
Linear classifiers are described geometrically by separating hyperplanes: an affine function assigns each point a signed deviation from the hyperplane, and the sign of that deviation determines the predicted class[12](https://web.stanford.edu/class/stats202/notes/Support-vector-machines/Support-vector-machines.html?highlight=e+learning). When two classes of data can be separated by a hyperplane, infinitely many hyperplanes separate them, and the support vector machine selects the one with the widest empty margin around it[12](https://web.stanford.edu/class/stats202/notes/Support-vector-machines/Support-vector-machines.html?highlight=e+learning).
In the linear support vector machine the hyperplane is written \( \mathbf{w}^T \mathbf{x} + b = 0 \), with the scale constraint \( \min_i |\mathbf{w}^T \mathbf{x}_i + b| = 1 \). This implies \( \mathbf{w}^T(\mathbf{x}_a - \mathbf{x}_b) = 2 \) for the closest pair of opposite-class points, so the margin is \( \rho = \|\mathbf{x}_a - \mathbf{x}_b\|_2 = 2/\|\mathbf{w}\|_2 \)[8](https://web.stanford.edu/class/cs276/handouts/lecture14-SVMs-handout-1-per.pdf). The training examples that fall exactly on the margin, with functional margin 1, are the support vectors, and they alone determine the decision function[8](https://web.stanford.edu/class/cs276/handouts/lecture14-SVMs-handout-1-per.pdf). When no hyperplane separates the classes cleanly, the support vector classifier relaxes the constraints by allowing per-point slack \( \epsilon_i \), with the total slack bounded by a budget parameter \( C \)[12](https://web.stanford.edu/class/stats202/notes/Support-vector-machines/Support-vector-machines.html?highlight=e+learning).
A key computational fact is that finding the hyperplane requires only dot products between pairs of input vectors. Replacing those dot products with a kernel function \( K(x_i, x_k) = \langle \Phi(x_i), \Phi(x_k) \rangle \) for a feature map \( \Phi \) yields the support vector machine, which fits a hyperplane in an expanded, possibly infinite-dimensional feature space while producing a nonlinear boundary in the original space[12](https://web.stanford.edu/class/stats202/notes/Support-vector-machines/Support-vector-machines.html?highlight=e+learning).
## References
1. [Hyperplane, MathWorld (Eric W. Weisstein)](https://sanweb.lib.msu.edu/crcmath/math/math/h/h453.htm)
2. [Hyperplane, Tangente Magazine geometry glossary](https://tangente-mag.com/en/articles/hyperplane)
3. [Plane, Wolfram MathWorld](https://mathworld.wolfram.com/Plane.html)
4. [Vector space, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Vector_space)
5. [Is a hyperplane uniquely defined by a single normal vector?, Mathematics Stack Exchange](https://math.stackexchange.com/questions/4478169/is-a-hyper-plane-uniquely-defined-by-a-single-normal-vector)
6. [Does every hyperplane have a unique normal vector (up to non-zero scalar multiplication)?, Mathematics Stack Exchange](https://math.stackexchange.com/questions/2796428/does-every-hyperplane-have-a-unique-normal-vector-up-to-non-zero-scalar-multipl)
7. [Linear variety, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Linear_variety)
8. [Linear Support Vector Machine, Stanford CS276 lecture handout (Introduction to Information Retrieval, Sec. 15.1)](https://web.stanford.edu/class/cs276/handouts/lecture14-SVMs-handout-1-per.pdf)
9. [Hyperplanes as Level Sets of Nonzero Linear Functionals, Androma theorem statement and proof](https://androma.org/theorems/13566)
10. [Hyperplane, nLab](https://ncatlab.org/nlab/show/hyperplane)
11. [Linear Algebra §9.5: Definition of Projective Spaces (Bourbaki algebra chapter notes), MIT](https://math.mit.edu/~hrm/palestine/bourbaki-algebra-1i.pdf)
12. [Support vector machines, Stanford STATS 202 course notes](https://web.stanford.edu/class/stats202/notes/Support-vector-machines/Support-vector-machines.html?highlight=e+learning)
13. [Arrangement of hyperplanes, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Arrangement_of_hyperplanes)
14. [Hyperplane Arrangements: Computations and Conjectures, arXiv:1101.0356](https://arxiv.org/abs/1101.0356)
15. [A New Algorithm for Enumeration of Cells of Hyperplane Arrangements and a Comparison with Avis and Fukuda's Reverse Search, SIAM Journal on Discrete Mathematics](https://epubs.siam.org/doi/10.1137/15M1027930)
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