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Parabola: pin string construction. The definition of a parabola by its focus and directrix can be used for drawing it with help of pins and strings: [9] Choose the focus and the directrix of the parabola. Take a triangle of a set square and prepare a string with length | | (see diagram).
String art, created with thread and paper A string art representing a projection of the 8-dimensional 4 21 polytope Quadratic Béziers in string art: The end points (•) and control point (×) define the quadratic Bézier curve (⋯). The arc is a segment of a parabola.
The definition of a hyperbola by its foci and its circular directrices (see above) can be used for drawing an arc of it with help of pins, a string and a ruler: [9] Choose the foci F 1 , F 2 {\displaystyle F_{1},F_{2}} and one of the circular directrices , for example c 2 {\displaystyle c_{2}} (circle with radius 2 a {\displaystyle 2a} )
An involute of a curve is the locus of a point on a piece of taut string as the string is either unwrapped from or wrapped around the curve. [1] The evolute of an involute is the original curve. It is generalized by the roulette family of curves. That is, the involutes of a curve are the roulettes of the curve generated by a straight line.
A parabola has as (two-sided) offsets rational curves of degree 6. A hyperbola or an ellipse has as (two-sided) offsets an algebraic curve of degree 8. A Bézier curve of degree n has as (two-sided) offsets algebraic curves of degree 4n − 2. In particular, a cubic Bézier curve has as (two-sided) offsets algebraic curves of degree 10.
A general straight-line thread connects the two points (0, k−t) and (t, 0), where k is an arbitrary scaling constant, and the family of lines is generated by varying the parameter t. From simple geometry, the equation of this straight line is y = −( k − t ) x / t + k − t .
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The string compass drawing tool has various modifications and designs, and this construction method is reminiscent of string-based methods for creating ellipses (with two fixed pins). Yet another mechanical method is a variant of the previous string compass method, providing greater precision and more flexibility.