By Wassenaar J.
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Additional info for 2D Curves
Till know we spoke about a single circle, but more circles together can be handy too: two circles we find in 8) the English bicycle , and the French bicyclette. Three arcs of a circle form together a circular triangle. In the case of three equal arcs beginning in the angles of an equal triangle, it is called the Reuleaux triangle. Already Archimedes described the arbelos or shoemaker's knife, also consisting of three circles. More concentric circles can be found at a target for shooting or darting.
10) A circle placed around a cross is the symbol of the Ku−klux klan movement. During the 19th century it was a popular occupation in geometry to construct all sorts of circles inside a triangle. Relatively easy to imagine are: • outside circle: circle through the vertices of the triangle • inside circle: largest circle that fits completely inside the triangle • touching circle: circle that touches an outer side of a triangle (two side are being lengthened) Less common circles are: • Feuerbach 's circle (1800−1834) or ninepointcircle It appears to be a fact that in each triangle the following points form a circle: footing points of the perpendiculars; middles of the sides; middles of the lines who connect the high point with the vertices.
An example of this relationship is formed by a pair of hyperbolas x y = 1 or x y = −1 Some relations with other curves: • the hyperbola is the isoptic of the parabola • the hyperbola can be seen as a sinusoidal spiral • depending on the center of inversion and the form of the hyperbola, as polar inverse the following curves can be obtained: ♦ lemniscate (of Bernoulli) (center) ♦ (right) strophoid (vertex) ♦ trisectrix of MacLaurin (vertex) • the pedal curve (focus) is the circle • a pedal (center) of the rectangular hyperbola is the lemniscate • its isoptic is the circle The Greek Euklid and Aristaeus wrote about the general hyperbola, although they confined the curve to one branch.