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Pure and applied geometry with Geometrica

Bruno Autin
Organization: MathSoft Overseas

2002 Applications of Computer Algebra Conference (ACA '2002)
Conference location

Volos, Greece

Geometrica is an application of Mathematica whose symbolic, numeric and graphical functions are used to create a package of 2D and 3D geometry. The functions cover theorems, constructions and drawings involving points, lines, segments, conics, planes, polyhedra, quadrics, curves and surfaces. They act on primitives which are the objects manipulated by Geometrica. The number of primitives is kept minimum. Points, lines, conics and planes are cartesian objects with operators CPoint, CLine, CConic and CPlane respectively. The capital letter C reminds the cartesian definition and avoids any confusion with similar names in Mathematica. Curves and surfaces are given by their parametric representation with one or two parameters and the generic operator is PPoint where the capital P stands for parametric. Segments, broken lines and polygons are considered as generalized segments and the operator is Segment. For polyhedra, the operator Solid acts on the faces of a polyhedron. The euclidean geometry fits the practical applications because of its great versatility. Polymorphic definitions reminded by capital E when necessary are then well adapted. A circle for instance can be defined by center and radius, of given center and passing through a point or of given center and tangent to a line. The function associated with a euclidean circle is ECircle[a,b] where a is a point and b either a number, a point or a line. Elaborate constructs may be derived from theorems (Brianchon, Desargues, Euler, Pascal, ...). Tests can be performed to make sure that points belong to some element, lines pass through a common point, lines or planes are parallel, figures are tangent, etc. The following transformations are treated: translation, rotation about point or line, symmetry about point, line and plane, inversion, similarity and projection either orthogonal or from a point or parallel to a vector. All the 2D objects can be intersected and, in 3D, intersections of any object with a plane is calculated analytically whenever possible or numerically otherwise. Distances, lengths and areas can be evaluated. Last the universal functions Draw and Draw3D can render legends and all the geometrical objects with a great variety of styles and colors. Geometrica can be called like any standard package of Mathematica and thus enrich the library of functions needed for a speial application. Optics is chosen to show how easy it may be to get with a remarkable simplicity results which usually require complicated coding. The examples concern the drawing of the closed orbits, the paraxial trajectories and the whole structure of a particle accelerator.

*Mathematics > Geometry
*Wolfram Technology > Application Packages > Applications from Independent Developers > Geometrica

2D and 3D geometry, theorems, constructions, segments, conics, planes, polyhedra, quadrics, Geometrica
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