Editing Turtle graphics (section)

==Extension to three dimensions==
[[File:Spheres-turtle.png|thumb|[[3D computer graphics|3D turtle graphics]] generated with [[Cheloniidae Turtle Graphics]]]]
[[File:Square pattern.png|thumb|left|Pattern drawn with a Python program]]
The ideas behind turtle graphics can be extended to include three-dimensional space. This is achieved by using one of several different coordinate models. A common setup is cartesian-rotational as with the original 2D turtle: an additional "up" vector ([[normal vector]]) is defined to choose the plane the turtle's 2D "forward" vector rotates in; the "up" vector itself also rotates around the "forward" vector. In effect, the turtle has two different heading angles, one within the plane and the other determining the plane's angle. Usually changing the plane's angle does not move the turtle, in line with the traditional setup.

Verhoeff 2010<ref>{{cite journal |last1=Verhoeff |first1=Tom |title=3D turtle geometry: artwork, theory, program equivalence and symmetry |journal=International Journal of Arts and Technology |date=2010 |volume=3 |issue=2/3 |pages=288–319 |doi=10.1504/IJART.2010.032569 |access-date=28 February 2021 |url=https://www.researchgate.net/publication/249923159}}</ref> implements the two vector approach; a ''roll'' command is used to rotate the "up" vector around the "forward" vector. The article proceeds to develop an algebraic theory to prove geometric properties from syntactic properties of the underlying turtle programs. One of the insights is that a ''dive'' command is really a shorthand of a turn-roll-turn sequence.

Cheloniidae Turtle Graphics is a 3D turtle library for [[Java (programming language)|Java]]. It has a ''bank'' command (same as ''roll'') and a ''pitch'' command (same as ''dive'') in the "Rotational Cartesian Turtle". Other coordinate models, including non-Euclidean geometry, are allowed but not included.<ref>[http://spencertipping.com/cheloniidae/ Spencer Tipping on cheloniidae](retrieved 2016-9-17)</ref>
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