Track point

Trace point is a term used in analytical and representational geometry , which refers to the intersection of lines and planes in three-dimensional space with the coordinate planes or axes . ${\displaystyle \mathbb {R} ^{3}}$

Track points of a straight line

Track points of a straight line

The points of intersection of the straight line with the coordinate planes are referred to as the track points of a straight line in three-dimensional space . The point at which the straight line penetrates the xy basic plane with the equation means that the track points and are defined analogously . For example, if a straight line equation is given in parametric form as follows ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle z=0}$ ${\displaystyle S_{xy}}$ ${\displaystyle S_{xz}}$ ${\displaystyle S_{yz}}$

${\displaystyle {\begin{pmatrix}x\\y\\z\end{pmatrix}}={\begin{pmatrix}3\\2\\-4\end{pmatrix}}+\lambda {\begin{pmatrix}4\\3\\-2\end{pmatrix}}}$with , ${\displaystyle \lambda \in \mathbb {R} }$

then the results from zeroing component: . The position vector of the tracking point is by inserting determined in the parametric representation: . The track point therefore has the coordinates . ${\displaystyle z}$ ${\displaystyle \lambda =-2}$ ${\displaystyle \lambda }$ ${\displaystyle {\vec {s}}_{xy}={\begin{pmatrix}-5\\-4\\0\end{pmatrix}}}$ ${\displaystyle S_{xy}=P(-5\mid -4\mid 0)}$

The prerequisite for the existence of a track point with a coordinate plane is that the straight line must not run parallel to this plane.

Track points of a plane

The track points of a plane in three-dimensional space are the points of intersection of the plane with the coordinate axes. Their designation is based on the coordinate axis that is intersected in each case. The calculation can be made from the intercept form or the coordinate form of a plane equation . ${\displaystyle \mathbb {R} ^{3}}$

For example, if the plane is given as follows in coordinate form: so is obtained by zeroing the - and component: . The track point thus has the coordinates . The two further track points can be determined accordingly. ${\displaystyle 6x+4y-3z=12}$ ${\displaystyle y}$ ${\displaystyle z}$ ${\displaystyle x=2}$ ${\displaystyle S_{x}=P(2\mid 0\mid 0)}$

The prerequisite for the existence of a track point with one of the coordinate axes is that it must not run parallel to one of the coordinate planes.

See also

• Straight track
• Track triangle

Web links

Wikibooks: Representation of track points with examples  - learning and teaching materials

• Trace points: tasks with solution tips. (PDF) In: Lower Saxony Education Server. Retrieved August 20, 2016 . 

Individual evidence

1. ^ Heinz Rapp: Mathematics for the technical college: Algebra, geometry, differential calculus, integral calculus, vector calculus, complex calculation . Springer Verlag, Heidelberg / Berlin 2010, ISBN 978-3-8348-0914-8 , pp. 451 ( limited preview in Google Book search).
2. ^ Institute of Computational Mathematics at the Technical University of Braunschweig: trace points and vanishing points. (PDF) In: Descriptive Geometry for Architects and Civil Engineers. Script and face-to-face exercises. WS 2010/11. P. 10 , accessed on August 20, 2016 . 
3. ↑ Jörg Stark: Training Intensive Mathematics: Analytical Geometry and Linear Algebra with learning videos online . Pons-Verlag , Stuttgart 2013, ISBN 978-3-12-949193-5 , pp. 37 ( limited preview in Google Book search).
4. ↑ Heinz Griesel u. a .: elements of mathematics. Qualification phase technology . Schroedel Verlag, Braunschweig 2013, ISBN 978-3-507-87034-5 , pp. 267 .
5. ↑ Cornelie Leopold: Geometric Basics of Architectural Representation . Springer Verlag, Heidelberg / Berlin 2011, ISBN 978-3-8348-1986-4 , p. 199 ( limited preview in Google Book search).