Parametric shape

The parametric form , or point towards the form is in mathematics a special form of a linear equation or plane equation . In the parametric form, a straight line is represented by a position vector (support vector) and a direction vector . Each point on the straight line is then described as a function of a parameter . A plane is represented by a support vector and two direction vectors. Each point on the plane is then described as a function of two parameters. The parametric form is therefore a special parametric representation .

Parametric form of a straight line equation

Parametric representation of a straight line

presentation

In the parametric form, a straight line in the plane is described by a support vector and a direction vector . A straight line then consists of those points in the plane whose position vectors give the equation ${\displaystyle {\vec {p}}}$ ${\displaystyle {\vec {u}}}$ ${\displaystyle {\vec {x}}}$

${\displaystyle {\vec {x}}={\vec {p}}+s{\vec {u}}}$   With   ${\displaystyle s\in \mathbb {R} }$

fulfill. The support vector is the position vector of any point on the straight line, which is also referred to as the starting point. The direction vector is the difference vector (connection vector) to any further point on the straight line. In the parametric form, the points of the straight lines are shown depending on the parameter . Each value of corresponds to exactly one point on the straight line. If the parameter runs through the real numbers, all points of the straight line are obtained. If a unit vector , then it specifies the distance of a point on the straight line from the starting point. ${\displaystyle s}$ ${\displaystyle s}$ ${\displaystyle {\vec {u}}}$ ${\displaystyle |s|}$

example

Written out is the parametric form of a straight line equation

${\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\end{pmatrix}}={\begin{pmatrix}p_{1}\\p_{2}\end{pmatrix}}+s{\begin{pmatrix}u_{1}\\u_{2}\end{pmatrix}}={\begin{pmatrix}p_{1}+su_{1}\\p_{2}+su_{2}\end{pmatrix}}}$

with . In the picture above is the support vector and the direction vector , one gets as a straight line equation ${\displaystyle s\in \mathbb {R} }$ ${\displaystyle {\vec {p}}={\tbinom {2}{2}}}$ ${\displaystyle {\vec {u}}={\tbinom {2}{-1}}}$

${\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\end{pmatrix}}={\begin{pmatrix}2\\2\end{pmatrix}}+s{\begin{pmatrix}2\\-1\end{pmatrix}}={\begin{pmatrix}2+2s\\2-s\end{pmatrix}}}$.

Each choice of , for example or , then results in a straight line point. ${\displaystyle s}$ ${\displaystyle s=0}$ ${\displaystyle s=1}$

calculation

From the two-point form

From the two-point form of a straight line equation, a direction vector of the straight line can be obtained as a difference vector between the position vectors and the two points, that is to say ${\displaystyle {\vec {p}}}$ ${\displaystyle {\vec {q}}}$

${\displaystyle {\vec {u}}={\vec {q}}-{\vec {p}}}$.

The position vector of one of the points can be used as a support vector . ${\displaystyle {\vec {p}}}$

From the normal form

A direction vector of the straight line can be determined from the normal form of a straight line equation by interchanging the two components of the normal vector of the straight line and changing the sign of one of the two components, that is to say ${\displaystyle {\vec {n}}}$

${\displaystyle {\vec {u}}={\begin{pmatrix}-n_{2}\\n_{1}\end{pmatrix}}}$.

The support vector can be taken from the normal form. ${\displaystyle {\vec {p}}}$

From the coordinate form

From the coordinate form of a straight line equation with the parameters and a normal vector of the straight line can be read off directly as and thus a direction vector of the straight line analogous to the normal form ${\displaystyle a,b}$ ${\displaystyle c}$ ${\displaystyle {\vec {n}}=(a,b)^{T}}$

${\displaystyle {\vec {u}}={\begin{pmatrix}-b\\a\end{pmatrix}}}$

determine. A support vector for the straight line is obtained by choosing whether or not it is zero ${\displaystyle a}$ ${\displaystyle b}$

${\displaystyle {\vec {p}}={\begin{pmatrix}c/a\\0\end{pmatrix}}}$   or   . ${\displaystyle {\vec {p}}={\begin{pmatrix}0\\c/b\end{pmatrix}}}$

In this way, a support vector and a direction vector can also be calculated from the axis intercept form and the Hessian normal form .

generalization

In general, the parametric shape can be used to describe not only straight lines in the plane, but also straight lines in three-dimensional or higher-dimensional space. In -dimensional Euclidean space , a straight line accordingly consists of those points whose position vectors give the equation ${\displaystyle n}$ ${\displaystyle {\vec {x}}}$

${\displaystyle {\vec {x}}={\vec {p}}+s{\vec {u}}}$   With   ${\displaystyle s\in \mathbb {R} }$

fulfill. It is only calculated with -component instead of two-component vectors. ${\displaystyle n}$

Parametric form of a plane equation

Parametric representation of a plane

presentation

In the parameter form a plane in three dimensional space by a support vector and two direction vectors and described. A plane then consists of those points in space whose position vectors give the equation ${\displaystyle {\vec {p}}}$ ${\displaystyle {\vec {u}}}$ ${\displaystyle {\vec {v}}}$ ${\displaystyle {\vec {x}}}$

${\displaystyle {\vec {x}}={\vec {p}}+s{\vec {u}}+t{\vec {v}}}$   With   ${\displaystyle s,t\in \mathbb {R} }$

fulfill. The support vector is the position vector of any point in the plane, which in turn is called the starting point. The two direction vectors, also called tension vectors here, must lie in the plane and not be equal to the zero vector . You may not collinear be, that is should not be a multiple of can write and vice versa. In the parametric form, the points of the plane are shown depending on the two parameters and . Exactly one point on the plane then corresponds to each value pair of these parameters. The direction vectors thus span an affine coordinate system, the affine coordinates being of a point on the plane. ${\displaystyle {\vec {u}}}$ ${\displaystyle {\vec {v}}}$ ${\displaystyle s}$ ${\displaystyle t}$ ${\displaystyle (s,t)}$

example

Written out is the parametric form of a plane equation

${\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}={\begin{pmatrix}p_{1}\\p_{2}\\p_{3}\end{pmatrix}}+s{\begin{pmatrix}u_{1}\\u_{2}\\u_{3}\end{pmatrix}}+t{\begin{pmatrix}v_{1}\\v_{2}\\v_{3}\end{pmatrix}}={\begin{pmatrix}p_{1}+su_{1}+tv_{1}\\p_{2}+su_{2}+tv_{2}\\p_{3}+su_{3}+tv_{3}\end{pmatrix}}}$

with . For example, if the support vector and if the directional vectors are and , then the plane equation is obtained ${\displaystyle s,t\in \mathbb {R} }$ ${\displaystyle {\vec {p}}=(3,2,1)^{T}}$ ${\displaystyle {\vec {u}}=(2,-1,0)^{T}}$ ${\displaystyle {\vec {v}}=(-1,0,2)^{T}}$

${\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}={\begin{pmatrix}3\\2\\1\end{pmatrix}}+s{\begin{pmatrix}2\\-1\\0\end{pmatrix}}+t{\begin{pmatrix}-1\\0\\2\end{pmatrix}}={\begin{pmatrix}3+2s-t\\2-s\\1+2t\end{pmatrix}}}$.

Each choice of , for example or , then results in a plane point. ${\displaystyle (s,t)}$ ${\displaystyle (0,0)}$ ${\displaystyle (1,2)}$

calculation

From the three-point form

From the three points form a plane equation two direction vectors of the plane can be the difference vectors between the position vectors , and points of each two get so ${\displaystyle {\vec {p}}}$ ${\displaystyle {\vec {q}}}$ ${\displaystyle {\vec {r}}}$

${\displaystyle {\vec {u}}={\vec {q}}-{\vec {p}}}$   and   . ${\displaystyle {\vec {v}}={\vec {r}}-{\vec {p}}}$

The position vector of one of the points can be used as a support vector . ${\displaystyle {\vec {p}}}$

From the normal form

From the normal form of a plane equation , two direction vectors of the plane can be derived from the normal vector by setting ${\displaystyle {\vec {n}}}$

${\displaystyle {\vec {u}}={\begin{pmatrix}-n_{2}\\n_{1}\\0\end{pmatrix}}}$   and   ${\displaystyle {\vec {v}}={\begin{pmatrix}0\\-n_{3}\\n_{2}\end{pmatrix}}}$

to be determined. If one of these two vectors is equal to the zero vector , the vector can be chosen instead. The support vector can be taken from the normal form. ${\displaystyle (-n_{3},0,n_{1})^{T}}$ ${\displaystyle {\vec {p}}}$

From the coordinate form

From the coordinate form of a plane equation with the parameters and a normal vector of the plane can be read off as, and thus two direction vectors of the plane over ${\displaystyle a,b,c}$ ${\displaystyle d}$ ${\displaystyle {\vec {n}}=(a,b,c)^{T}}$

${\displaystyle {\vec {u}}={\begin{pmatrix}-b\\a\\0\end{pmatrix}}}$   and   ${\displaystyle {\vec {v}}={\begin{pmatrix}0\\-c\\b\end{pmatrix}}}$

determine. If one of these two vectors is equal to the zero vector, the vector can be chosen instead. A support vector is obtained, depending on which of the numbers is not equal to zero, by choosing ${\displaystyle (-c,0,a)^{T}}$ ${\displaystyle a,b,c}$

${\displaystyle {\vec {p}}={\begin{pmatrix}d/a\\0\\0\end{pmatrix}},~{\vec {p}}={\begin{pmatrix}0\\d/b\\0\end{pmatrix}}}$   or   . ${\displaystyle {\vec {p}}={\begin{pmatrix}0\\0\\d/c\end{pmatrix}}}$

Similarly, a support vector and one or two direction vectors can be calculated in this way from the axis intercept form and the Hessian normal form .

generalization

In general, the parameter shape can be used to describe not only planes in three-dimensional space, but also in higher-dimensional spaces. In -dimensional Euclidean space, a plane accordingly consists of those points whose position vectors give the equation ${\displaystyle n}$ ${\displaystyle {\vec {x}}}$

${\displaystyle {\vec {x}}={\vec {p}}+s{\vec {u}}+t{\vec {v}}}$   With   ${\displaystyle s,t\in \mathbb {R} }$

fulfill. It is only calculated with -component instead of three-component vectors. ${\displaystyle n}$

See also

• Parametric representation

literature

• Steffen Goebbels, Stefan Ritter: Understanding and applying mathematics . Springer, 2011, ISBN 978-3-8274-2762-5 .

Web links

• Convert plane from normal form to parametric form. In: Serlo. Retrieved February 23, 2014 . 
• Convert plane from coordinate form to parametric form. In: Serlo. Retrieved February 23, 2014 .