Real plane : Linear algebra

Linear algebra (also known as vector algebra) is a branch of mathematics that deals with vector spaces. Similar to other branches of mathematics, structure-preserving mappings, which in linear algebra are linear mappings, are of particular interest. CGRAPH allows the representation of straight lines and circles in 2D space and can calculate intersections.

A straight line in parametric representation has a starting point and a direction vector. Both can be entered separately for the x and y coordinates in the Straight lines/Parametric form tab. Clicking on the Add to list button then displays the straight line in the right-hand window. A check is performed to see whether the straight line is already part of the list. CGRAPH also recognises entries that do not have exactly the same definition (e.g. a different starting point) but still describe the same straight line.

A line in normal vector representation has a starting point and a vector that is perpendicular to the line. Both can be entered separately for the x and y coordinates in the Lines/Normal Form tab. Clicking on the Add to list button then displays the line in the right-hand window. A check is performed to see whether the line is already part of the list. CGRAPH also recognises entries that do not have exactly the same definition (e.g. a different starting point) but still describe the same line.

A circle is described by the absolute value of the difference between a vector (x/y) and the centre of the circle. For all (x/y) on the circle, this absolute value is equal to the radius of the circle. The radius and centre (separately for x and y coordinates) can be entered in the Circles tab. Clicking on the Add to list button then displays the circle in the right-hand window. A check is made to see whether the circle is already part of the list.

A point is described by its position vector (x/y). The position vector can be entered (separately for x and y coordinates) in the Points tab. Clicking on the Add to list button then displays the point in the right-hand window. A check is performed to see whether the point is already part of the list.

The Intersections function also uses points in this format. The intersections determined and saved there can therefore be loaded from a file and displayed again here.

After clicking on the Create graphic button, CGRAPH switches to the Display tab and displays the graphic. However, the graphics are created in any case, as no calculation is required.

The default values can be viewed in the tree on the left-hand side. Clicking on the coloured symbol for the graph allows you to quickly hide the graphic from the display. Double-clicking on the text allows you to adjust the general colour.

The object list allows you to export and import entire object lists. In addition, individual entries can be deleted by selecting them.

The individual objects can also be displayed independently of each other. While only one basic colour can be selected in the tree view, different colours or displays can be assigned to each individual object in the list. Double-clicking on the respective object opens a settings dialogue where the object can be modified. Note: if the colour of an object has been changed in this way, it can no longer be modified using the general colour selection in the tree view.

Notes on operating the graphic display

The Determine intersections function can be used later to determine the intersections of the objects with each other. Unlike the other graphics, these are not determined graphically, but are actually calculated based on the values entered.

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