Coplanarity

Last updated November 07, 2023

In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all. For example, three points are always coplanar, and if the points are distinct and non-collinear, the plane they determine is unique. However, a set of four or more distinct points will, in general, not lie in a single plane.

Properties in three dimensions

In three-dimensional space, two linearly independent vectors with the same initial point determine a plane through that point. Their cross product is a normal vector to that plane, and any vector orthogonal to this cross product through the initial point will lie in the plane.^[1] This leads to the following coplanarity test using a scalar triple product:

Four distinct points, $x 1, x 2, x 3, x 4$ , are coplanar if and only if,

[(x_{2}-x_{1})\times (x_{4}-x_{1})]\cdot (x_{3}-x_{1})=0.

which is also equivalent to

(x_{2}-x_{1})\cdot [(x_{4}-x_{1})\times (x_{3}-x_{1})]=0.

If three vectors $a, b, c$ are coplanar, then if $a \cdot b = 0$ (i.e., $a$ and $b$ are orthogonal) then

(\mathbf {c} \cdot \mathbf {\hat {a}} )\mathbf {\hat {a}} +(\mathbf {c} \cdot \mathbf {\hat {b}} )\mathbf {\hat {b}} =\mathbf {c} ,

where $\mathbf {\hat {a}}$ denotes the unit vector in the direction of $a$ . That is, the vector projections of $c$ on $a$ and $c$ on $b$ add to give the original $c$ .

Coplanarity of points in n dimensions whose coordinates are given

Since three or fewer points are always coplanar, the problem of determining when a set of points are coplanar is generally of interest only when there are at least four points involved. In the case that there are exactly four points, several ad hoc methods can be employed, but a general method that works for any number of points uses vector methods and the property that a plane is determined by two linearly independent vectors.

In an $n$ -dimensional space where $n \geq 3$ , a set of $k$ points $\{p_{0},\ p_{1},\ \dots ,\ p_{k-1}\}$ are coplanar if and only if the matrix of their relative differences, that is, the matrix whose columns (or rows) are the vectors ${\overrightarrow {p_{0}p_{1}}},\ {\overrightarrow {p_{0}p_{2}}},\ \dots ,\ {\overrightarrow {p_{0}p_{k-1}}}$ is of rank 2 or less.

For example, given four points

{\begin{aligned}X&=(x_{1},x_{2},\dots ,x_{n}),\\Y&=(y_{1},y_{2},\dots ,y_{n}),\\Z&=(z_{1},z_{2},\dots ,z_{n}),\\W&=(w_{1},w_{2},\dots ,w_{n}),\end{aligned}}

if the matrix

{\begin{bmatrix}x_{1}-w_{1}&x_{2}-w_{2}&\dots &x_{n}-w_{n}\\y_{1}-w_{1}&y_{2}-w_{2}&\dots &y_{n}-w_{n}\\z_{1}-w_{1}&z_{2}-w_{2}&\dots &z_{n}-w_{n}\\\end{bmatrix}}

is of rank 2 or less, the four points are coplanar.

In the special case of a plane that contains the origin, the property can be simplified in the following way: A set of $k$ points and the origin are coplanar if and only if the matrix of the coordinates of the $k$ points is of rank 2 or less.

Geometric shapes

A skew polygon is a polygon whose vertices are not coplanar. Such a polygon must have at least four vertices; there are no skew triangles.

A polyhedron that has positive volume has vertices that are not all coplanar.

Related Research Articles

In mathematics, physics, and engineering, a Euclidean vector or simply a vector is a geometric object that has magnitude and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a directed line segment, or graphically as an arrow connecting an initial pointA with a terminal pointB, and denoted by $.$

In Euclidean geometry, an affine transformation or affinity is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.

Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies without considering the forces that cause them to move. Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of mathematics. A kinematics problem begins by describing the geometry of the system and declaring the initial conditions of any known values of position, velocity and/or acceleration of points within the system. Then, using arguments from geometry, the position, velocity and acceleration of any unknown parts of the system can be determined. The study of how forces act on bodies falls within kinetics, not kinematics. For further details, see analytical dynamics.

In mathematics, the cross product or vector product is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space, and is denoted by the symbol $. Given two linearly independent vectors a and b, the cross product, a \times b, is a vector that is perpendicular to both a and b, and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product.$

<span class="mw-page-title-main">Moment of inertia</span> Scalar measure of the rotational inertia with respect to a fixed axis of rotation

The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis, akin to how mass determines the force needed for a desired acceleration. It depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rate of rotation by a given amount.

In geometry, a normal is an object that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at the point. A normal vector may have length one or its length may represent the curvature of the object. Multiplying a normal vector by -1 results in the opposite vector, which may be used for indicating sides.

In physics, the reciprocal lattice emerges from the Fourier transform of another lattice. The direct lattice or real lattice is a periodic function in physical space, such as a crystal system. The reciprocal lattice exists in the mathematical space of spatial frequencies, known as reciprocal space or k space, where $refers to the wavevector.$

In linear algebra, linear transformations can be represented by matrices. If $is a linear transformation mapping to and is a column vector with entries, then$

In geometry, a barycentric coordinate system is a coordinate system in which the location of a point is specified by reference to a simplex. The barycentric coordinates of a point can be interpreted as masses placed at the vertices of the simplex, such that the point is the center of mass of these masses. These masses can be zero or negative; they are all positive if and only if the point is inside the simplex.

In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector-valued vector triple product.

In geometry, Plücker coordinates, introduced by Julius Plücker in the 19th century, are a way to assign six homogeneous coordinates to each line in projective 3-space, $. Because they satisfy a quadratic constraint, they establish a one-to-one correspondence between the 4-dimensional space of lines in and points on a quadric in . A predecessor and special case of Grassmann coordinates, Plücker coordinates arise naturally in geometric algebra. They have proved useful for computer graphics, and also can be extended to coordinates for the screws and wrenches in the theory of kinematics used for robot control.$

Screw theory is the algebraic calculation of pairs of vectors, such as forces and moments or angular and linear velocity, that arise in the kinematics and dynamics of rigid bodies. The mathematical framework was developed by Sir Robert Stawell Ball in 1876 for application in kinematics and statics of mechanisms.

In mathematics, the real coordinate space of dimension $n$ , denoted $R n$ or $,$ is the set of the $n$ -tuples of real numbers, that is the set of all sequences of $n$ real numbers. Special cases are called the real line $R 1$ and the real coordinate plane $R 2$ . With component-wise addition and scalar multiplication, it is a real vector space, and its elements are called coordinate vectors.

The vector projection of a vector $a$ on a nonzero vector $b$ is the orthogonal projection of $a$ onto a straight line parallel to $b$ . The projection of $a$ onto $b$ is often written as $or a ∥ b .$

In three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel. A simple example of a pair of skew lines is the pair of lines through opposite edges of a regular tetrahedron. Two lines that both lie in the same plane must either cross each other or be parallel, so skew lines can exist only in three or more dimensions. Two lines are skew if and only if they are not coplanar.

In Euclidean geometry, the Fermat point of a triangle, also called the Torricelli point or Fermat–Torricelli point, is a point such that the sum of the three distances from each of the three vertices of the triangle to the point is the smallest possible or, equivalently, the geometric median of the three vertices. It is so named because this problem was first raised by Fermat in a private letter to Evangelista Torricelli, who solved it.

In geometry, a three-dimensional space is a mathematical space in which three values (coordinates) are required to determine the position of a point. Most commonly, it is the three-dimensional Euclidean space, that is, the Euclidean space of dimension three, which models physical space. More general three-dimensional spaces are called 3-manifolds. The term may also refer colloquially to a subset of space, a three-dimensional region, a solid figure.

In geometry, various formalisms exist to express a rotation in three dimensions as a mathematical transformation. In physics, this concept is applied to classical mechanics where rotational kinematics is the science of quantitative description of a purely rotational motion. The orientation of an object at a given instant is described with the same tools, as it is defined as an imaginary rotation from a reference placement in space, rather than an actually observed rotation from a previous placement in space.

In Euclidean space, the distance from a point to a plane is the distance between a given point and its orthogonal projection on the plane, the perpendicular distance to the nearest point on the plane.

In mathematics, specifically multilinear algebra, a dyadic or dyadic tensor is a second order tensor, written in a notation that fits in with vector algebra.

References

↑ Swokowski, Earl W. (1983), Calculus with Analytic Geometry (Alternate ed.), Prindle, Weber & Schmidt, p. 647, ISBN 0-87150-341-7

External links

Weisstein, Eric W. "Coplanar". MathWorld .

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Swokowski, Earl W. (1983), Calculus with Analytic Geometry (Alternate ed.), Prindle, Weber & Schmidt, p. 647, ISBN 0-87150-341-7

[1]