In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, an intersection is a point, line, or curve common to two or more objects (such as lines, curves, planes, and surfaces). The simplest case in
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
is the
line–line intersection
In Euclidean geometry, the intersection of a line and a line can be the empty set, a point (geometry), point, or another Line (geometry), line. Distinguishing these cases and finding the Intersection (Euclidean geometry), intersection have uses, ...
between two distinct
lines
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Arts ...
, which either is one
point
Point or points may refer to:
Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland
* Point ...
or does not exist (if the lines are
parallel
Parallel is a geometric term of location which may refer to:
Computing
* Parallel algorithm
* Parallel computing
* Parallel metaheuristic
* Parallel (software), a UNIX utility for running programs in parallel
* Parallel Sysplex, a cluster of IBM ...
). Other types of geometric intersection include:
*
Line–plane intersection
In analytic geometry, the intersection of a line and a plane in three-dimensional space can be the empty set, a point, or a line. It is the entire line if that line is embedded in the plane, and is the empty set if the line is parallel to the pl ...
*
Line–sphere intersection
In analytic geometry, a line and a sphere can intersect in three ways:
# No intersection at all
# Intersection in exactly one point
# Intersection in two points.
Methods for distinguishing these cases, and determining the coordinates for the ...
*
Intersection of a polyhedron with a line
*
Line segment intersection
In geometry, an intersection is a point, line, or curve common to two or more objects (such as lines, curves, planes, and surfaces). The simplest case in Euclidean geometry is the line–line intersection between two distinct lines, which eithe ...
*
Intersection curve
In geometry, an intersection curve is a curve that is common to two geometric objects. In the simplest case, the intersection of two non-parallel planes in Euclidean 3-space is a line (geometry), line. In general, an intersection curve consists of ...
Determination of the intersection of
flats
Flat or flats may refer to:
Architecture
* Flat (housing), an apartment in the United Kingdom, Ireland, Australia and other Commonwealth countries
Arts and entertainment
* Flat (music), a symbol () which denotes a lower pitch
* Flat (soldier), ...
– linear geometric objects embedded in a higher-
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
al space – is a simple task of
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices.
...
, namely the solution of a
system of linear equations
In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variable (math), variables.
For example,
:\begin
3x+2y-z=1\\
2x-2y+4z=-2\\
-x+\fracy-z=0
\end
is a system of three ...
. In general the determination of an intersection leads to
non-linear equation
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...
s, which can be
solved numerically, for example using
Newton iteration
In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valu ...
. Intersection problems between a line and a
conic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...
(circle, ellipse, parabola, etc.) or a
quadric
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is de ...
(sphere, cylinder, hyperboloid, etc.) lead to
quadratic equation
In algebra, a quadratic equation () is any equation that can be rearranged in standard form as
ax^2 + bx + c = 0\,,
where represents an unknown (mathematics), unknown value, and , , and represent known numbers, where . (If and then the equati ...
s that can be easily solved. Intersections between quadrics lead to
quartic equation
In mathematics, a quartic equation is one which can be expressed as a ''quartic function'' equaling zero. The general form of a quartic equation is
:ax^4+bx^3+cx^2+dx+e=0 \,
where ''a'' ≠ 0.
The quartic is the highest order polynomi ...
s that can be solved
algebraically.
On a plane
Two lines
For the determination of the intersection point of two non-parallel lines
one gets, from
Cramer's rule
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants o ...
or by substituting out a variable, the coordinates of the intersection point
:
:
(If
the lines are parallel and these formulas cannot be used because they involve dividing by 0.)
Two line segments
For two non-parallel
line segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
s
and
there is not necessarily an intersection point (see diagram), because the intersection point
of the corresponding lines need not to be contained in the line segments. In order to check the situation one uses parametric representations of the lines:
:
:
The line segments intersect only in a common point
of the corresponding lines if the corresponding parameters
fulfill the condition
.
The parameters
are the solution of the linear system
:
:
It can be solved for ''s'' and ''t'' using Cramer's rule (see
above). If the condition
is fulfilled one inserts
or
into the corresponding parametric representation and gets the intersection point
.
''Example:'' For the line segments
and
one gets the linear system
:
:
and
. That means: the lines intersect at point
.
''Remark:'' Considering lines, instead of segments, determined by pairs of points, each condition
can be dropped and the method yields the intersection point of the lines (see
above).
A line and a circle
For the intersection of
*line
and
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
one solves the line equation for or and
substitutes it into the equation of the circle and gets for the solution (using the formula of a quadratic equation)
with
:
:
if
If this condition holds with strict inequality, there are two intersection points; in this case the line is called a
secant line
Secant is a term in mathematics derived from the Latin ''secare'' ("to cut"). It may refer to:
* a secant line, in geometry
* the secant variety, in algebraic geometry
* secant (trigonometry) (Latin: secans), the multiplicative inverse (or reciproc ...
of the circle, and the line segment connecting the intersection points is called a
chord of the circle.
If
holds, there exists only one intersection point and the line is tangent to the circle. If the weak inequality does not hold, the line does not intersect the circle.
If the circle's midpoint is not the origin, see. The intersection of a line and a parabola or hyperbola may be treated analogously.
Two circles
The determination of the intersection points of two circles
*
can be reduced to the previous case of intersecting a line and a circle. By subtraction of the two given equations one gets the line equation:
:
This special line is the
radical line of the two circles.
Special case
:
In this case the origin is the center of the first circle and the second center lies on the x-axis (s. diagram). The equation of the radical line simplifies to
and the points of intersection can be written as
with
:
In case of