Tripod (geometry)

Three leg ( English trihedron ) referred to in the geometry of a geometric figure of the Euclidean space or the Euclidean plane , which consists of a common point and three outgoing from this point routes , or vectors of the same length is. It is generally assumed here that these lines or vectors do not all lie on a straight line . ${\displaystyle d>0}$

Definitions and explanations

• Formally, a tripod can be understood as a quadruple with three ( different in pairs ) stretches of the , which are all of the same length , have a common corner point and otherwise have no other common point. ${\displaystyle {\mathcal {D}}=(O,S_{1},S_{2},S_{3})}$ ${\displaystyle S_{1},S_{2},S_{3}}$ ${\displaystyle \mathbb {R} ^{n}\;(n\geq 2)}$ ${\displaystyle d>0}$ ${\displaystyle O\in \mathbb {R} ^{n}}$
• The point is called the vertex of the tripod . ${\displaystyle O}$ ${\displaystyle {\mathcal {D}}}$
• In particular, the four corner points of a tripod are not collinear and the corner points given next to the vertex do not coincide with the point . ${\displaystyle O}$ ${\displaystyle P_{i}\;(i=1,2,3)}$ ${\displaystyle O}$
• So for the tripod . The vector belonging to the route is usually identified with it. ${\displaystyle {\mathcal {D}}=(O,S_{1},S_{2},S_{3})}$ ${\displaystyle S_{i}=[O,P_{i}]}$ ${\displaystyle S_{i}}$ ${\displaystyle {\vec {v}}_{i}={\overrightarrow {O{P_{i}}}}}$
• The three vectors are twos - linearly independent and it applies . ${\displaystyle {\vec {v}}_{i}={\overrightarrow {O{P_{i}}}}\;(i=1,2,3)}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle d=\|{\vec {v}}_{i}\|_{2}=|S_{i}|\;(i=1,2,3)}$
• The associated quadruple is also called a tripod. ${\displaystyle {\mathcal {D}}^{'}=(O,{\vec {v}}_{1},{\vec {v}}_{2},{\vec {v}}_{3})}$
• A tripod is also used in relation to the corresponding point quadruple . ${\displaystyle {\mathcal {D}}^{''}=(O,P_{1},P_{2},P_{3})}$
• Usually, no further distinction is made between and the connection is taken for granted. ${\displaystyle {\mathcal {D}},{\mathcal {D}}^{'},{\mathcal {D}}^{''}}$

particularities

• If lines are in the Euclidean plane or in a plane of Euclidean space, they are called a plane tripod . ${\displaystyle S_{1},S_{2},S_{3}}$ ${\displaystyle {\mathcal {D}}}$
• If stretches in Euclidean space are not all in one plane, one calls a three-legged tripod . This is the case if and only if the vector is triple - linearly independent . ${\displaystyle S_{1},S_{2},S_{3}}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle ({\vec {v}}_{1},{\vec {v}}_{2},{\vec {v}}_{3})}$ ${\displaystyle \mathbb {R} }$
• If a three-dimensional tripod is in pairs and are perpendicular to one another , it is called an orthogonal three-dimensional tripod . ${\displaystyle {\mathcal {D}}}$ ${\displaystyle S_{1},S_{2},S_{3}}$ ${\displaystyle {\mathcal {D}}}$
• Is an orthogonal spatial tripod with , it is called an orthonormal spatial tripod . In this case, the apex is a cornerstone of the by spanned cube of side length . An orthonormal spatial tripod is therefore sometimes called a cube corner . ${\displaystyle {\mathcal {D}}}$ ${\displaystyle d=1}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle O}$ ${\displaystyle S_{1},S_{2},S_{3}}$ ${\displaystyle d=1}$
• As a rule, orthonormal spatial tripods appear in Euclidean intuition . If there is one, then the associated vector triple forms an orthonormal basis of . ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle ({\vec {v}}_{1},{\vec {v}}_{2},{\vec {v}}_{3})}$ ${\displaystyle \mathbb {R} ^{3}}$
• Tripods appear not least in descriptive geometry in connection with the fundamental theorem of axonometry . An image of an orthonormal spatial tripod created by parallel projection is referred to as a Pohlke's tripod .
• In differential geometry , a tripod is usually found as an accompanying tripod ( English moving trihedron or moving frame ) of a space curve , especially in connection with the Frenet formulas . Accompanying tripods are created when the quadruple is formed for each curve point of a space curve from itself, the adjacent tangent unit vector , the adjacent normal unit vector and the adjacent binormal unit vector . The vector triple always forms a legal system . ${\displaystyle c(s)\;(s\in [a,b])}$ ${\displaystyle c\colon [a,b]\to \mathbb {R} ^{3}}$ ${\displaystyle {\hat {t}}(s)}$ ${\displaystyle {\hat {n}}(s)}$ ${\displaystyle {\hat {b}}(s)}$ ${\displaystyle (c(s),{\hat {t}}(s),{\hat {n}}(s),{\hat {b}}(s))}$ ${\displaystyle ({\hat {t}}(s),{\hat {n}}(s),{\hat {b}}(s))}$
• In differential geometry, the accompanying (Frenet) n-legs are examined in generalization of the accompanying tripods to general space curves . ${\displaystyle c\colon [a,b]\to \mathbb {R} ^{n}\;(n\in \mathbb {N} ,n\geq 2)}$

Demarcation

The used in English for a tripod designation trihedron suggests a tripod with a Trieder equate, so with one of three flat surfaces bounded polyhedron in . In the German-language mathematical specialist literature, this equation is not generally used. If you follow György Hajós and his presentation in the Introduction to Geometry , then a Trieder is a special, unrestricted geometric figure of the . Hajós describes this as a triangular convex corner or as a three-sided infinite pyramid or briefly as a triangle and thus means the convex shell of three rays emanating from a common point in space that have no other point in space in common. He mentions the octants of the as examples of such trieder . ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle O\in \mathbb {R} ^{3}}$ ${\displaystyle O}$ ${\displaystyle \mathbb {R} ^{3}}$

literature

• Hermann Athens, Jörn Bruhn (ed.): Lexicon of school mathematics . and adjacent areas. tape 1 . AE. Aulis Verlag Deubner, Cologne 1976, ISBN 3-7614-0242-2 .
• Hermann Engesser (editor): The little Duden. Math . 2nd Edition. Dudenverlag, Mannheim / Leipzig / Vienna / Zurich 1996, ISBN 3-411-05352-6 .
• Walter Gellert, H. Küstner, M. Hellwich, H. Kästner (eds.): Small encyclopedia of mathematics . Verlag Harri Deutsch, Thun / Frankfurt / Main 197, ISBN 3-87144-323-9 .
• György Hajós : Introduction to Geometry . BSB BG Teubner Verlagsgesellschaft, Leipzig 1970 (Hungarian: Bevezetés A Geometriába . Translated by G. Eisenreich [Leipzig, also editor]).
• Wilhelm Klingenberg : A lecture on differential geometry (=  Heidelberg pocket books . Volume 107 ). Springer Verlag, Berlin (among others) 1973, ISBN 3-540-06253-X .
• John McCleary: Geometry from a Differentiable Viewpoint . Cambridge University Press, Cambridge (et al.) 2013, ISBN 978-0-521-13311-1 .
• Fritz Reinhardt, Heinrich Soeder (Ed.): Dtv-Atlas for Mathematics. Boards and texts . Volume 2: Analysis and Applied Mathematics. 11th revised and corrected edition. Deutscher Taschenbuch Verlag, Munich 2003, ISBN 3-423-03008-9 .

Web links

• Moving frame in the English language Wikipedia

Individual evidence

1. ^ György Hajós: Introduction to Geometry. 1970, pp. 244-245