Two-point form

The two -point form or two-point form is a special form of a straight line equation in mathematics . In the two-point form, a straight line is represented in the Euclidean plane or in Euclidean space with the help of two points of the straight line. The coordinate representation of a straight line in the plane takes place in the two-point form with the help of the gradient triangle of the straight line. In the vector representation, the position vector of one of the two points serves as a support vector of the straight line, while the difference vector to the position vector of the other point forms the direction vector of the straight line.

The form of a plane equation corresponding to the two-point form is called the three-point form.

Coordinate representation

presentation

Two-point form of a straight line equation

In the two-point form, a straight line in the plane that passes through the two different points and is described as the set of those points whose coordinates match the equation ${\displaystyle (x_{1},y_{1})}$ ${\displaystyle (x_{2},y_{2})}$ ${\displaystyle (x,y)}$

${\displaystyle {\frac {y-y_{1}}{x-x_{1}}}={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}}$

fulfill. Here and must be different and may not be chosen the same . If the straight line equation is solved, the explicit representation is obtained ${\displaystyle x_{1}}$ ${\displaystyle x_{2}}$ ${\displaystyle x}$ ${\displaystyle x_{1}}$ ${\displaystyle y}$

${\displaystyle y={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}\cdot (x-x_{1})+y_{1}}$,

which can also be used for. The representation is valid without restriction ${\displaystyle x=x_{1}}$

${\displaystyle (y-y_{1})\cdot (x_{2}-x_{1})=(x-x_{1})\cdot (y_{2}-y_{1})}$.

example

If, for example, the two given straight line points are and , then the straight line equation is obtained ${\displaystyle (1,3)}$ ${\displaystyle (3,2)}$

${\displaystyle {\frac {y-3}{x-1}}={\frac {2-3}{3-1}}}$

or resolved after ${\displaystyle y}$

${\displaystyle y={\frac {2-3}{3-1}}\cdot (x-1)+3=-{\frac {x}{2}}+{\frac {7}{2}}}$

respectively

${\displaystyle (y-3)\cdot (3-1)=(x-1)\cdot (2-3)}$.

Derivation

This representation of a straight line equation follows from the fact that for the slope of a straight line ${\displaystyle m}$

${\displaystyle m={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}}$

applies. According to the theorem of rays , any point on a straight line can be selected instead of the point without changing the ratio . This then also applies ${\displaystyle (x_{2},y_{2})}$ ${\displaystyle (x,y)}$ ${\displaystyle m}$

${\displaystyle m={\frac {y-y_{1}}{x-x_{1}}}}$.

By equating these two equations, the two-point form then follows. The latter equation corresponds to the point slope form of a straight line equation.

Representation as a determinant

A straight line that runs through two specified points can also be calculated using the determinant of a matrix using the equation

${\displaystyle \det {\begin{pmatrix}x&x_{1}&x_{2}\\y&y_{1}&y_{2}\\1&1&1\end{pmatrix}}=0}$

or equivalent to it

${\displaystyle \det {\begin{pmatrix}x-x_{1}&x_{2}-x_{1}\\y-y_{1}&y_{2}-y_{1}\end{pmatrix}}=0}$

To be defined. Such a representation is also called the determinant form of a straight line equation.

Vector illustration

Two-point form of a straight line equation with vectors

presentation

In vector representation, a straight line in the plane is described in the two-point form by the position vectors and two points of the straight line. A straight line then consists of those points in the plane whose position vectors give the equation ${\displaystyle {\vec {p}}}$ ${\displaystyle {\vec {q}}}$ ${\displaystyle {\vec {x}}}$

${\displaystyle {\vec {x}}={\vec {p}}+t({\vec {q}}-{\vec {p}})}$   For   ${\displaystyle t\in \mathbb {R} }$

fulfill. The vector serves as a support vector for the straight line, while the difference vector forms the direction vector of the straight line. The points of the straight line are shown as a function of the parameter , with exactly one point of the straight line corresponding to each parameter value. This is a special parametric representation of the straight line. ${\displaystyle {\vec {p}}}$ ${\displaystyle {\vec {q}}-{\vec {p}}}$ ${\displaystyle t}$

example

Written out is the two-point form of a straight line equation

${\displaystyle {\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}p_{x}\\p_{y}\end{pmatrix}}+t{\begin{pmatrix}q_{x}-p_{x}\\q_{y}-p_{y}\end{pmatrix}}={\begin{pmatrix}p_{x}+t(q_{x}-p_{x})\\p_{y}+t(q_{y}-p_{y})\end{pmatrix}}}$

with . If, for example, the two position vectors are and , then the straight line equation is obtained ${\displaystyle t\in \mathbb {R} }$ ${\displaystyle {\vec {p}}={\tbinom {2}{2}}}$ ${\displaystyle {\vec {q}}={\tbinom {4}{1}}}$

${\displaystyle {\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}2\\2\end{pmatrix}}+t{\begin{pmatrix}4-2\\1-2\end{pmatrix}}={\begin{pmatrix}2+2t\\2-t\end{pmatrix}}}$.

Each choice of , for example or , then results in a straight line point. ${\displaystyle t}$ ${\displaystyle t=2}$ ${\displaystyle t=-1}$

calculation

From the parametric form of a straight line equation with support vector and direction vector , in addition to the support vector, a further position vector of a point on the straight line can be determined simply by selecting ${\displaystyle {\vec {p}}}$ ${\displaystyle {\vec {u}}}$

${\displaystyle {\vec {q}}={\vec {p}}+{\vec {u}}}$

Find. From the other forms of straight line equations, the coordinate form , the intercept form , the normal form and the Hessian normal form , the associated parametric form of the straight line is first determined (see calculation of the parametric form ) and then the two-point form.

Homogeneous coordinates

A related representation of a straight line using two straight line points uses barycentric coordinates . A straight line in the plane is then given by the equation

${\displaystyle {\vec {x}}=\lambda {\vec {p}}+\mu {\vec {q}}}$   for     with   ${\displaystyle \lambda ,\mu \in \mathbb {R} }$ ${\displaystyle \lambda +\mu =1}$

described. Here are the normalized barycentric coordinates of a straight line point. If both coordinates are positive, the straight line point lies between the two specified points; if one coordinate is negative, it is outside. The barycentric coordinates are special homogeneous affine coordinates , while inhomogeneous affine coordinates are used in the two-point form. ${\displaystyle (\lambda ,\mu )}$

generalization

In general, the two-point shape not only describes straight lines in the plane, but also in three- and higher-dimensional spaces. In -dimensional Euclidean space, a straight line accordingly consists of those points whose position vectors give the equation ${\displaystyle n}$ ${\displaystyle {\vec {x}}}$

${\displaystyle {\vec {x}}={\vec {p}}+t({\vec {q}}-{\vec {p}})}$   For   ${\displaystyle t\in \mathbb {R} }$

fulfill. It is only calculated with -component instead of two-component vectors. The representation with barycentric coordinates is also retained in analog form in higher-dimensional spaces. ${\displaystyle n}$

literature

• Lothar Papula: Mathematics for Engineers and Natural Scientists 1 . Springer, 2007, ISBN 978-3-8348-0224-8 .
• Thomas Westermann: Mathematics for Engineers . Springer, 2008, ISBN 978-3-540-77731-1 .