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Turtle graphics are often associated with the Logo programming language. [2] Seymour Papert added support for turtle graphics to Logo in the late 1960s to support his version of the turtle robot, a simple robot controlled from the user's workstation that is designed to carry out the drawing functions assigned to it using a small retractable pen set into or attached to the robot's body.
The trigonometric functions cosine and sine of angle θ may be defined on the unit circle as follows: If (x, y) is a point on the unit circle, and if the ray from the origin (0, 0) to (x, y) makes an angle θ from the positive x-axis, (where counterclockwise turning is positive), then = =.
It is also possible to bind a point to the inside of a (3D) circle or polygon. This feature, inherited from C.a.R., is based on barycentric coordinates. Since the 4.1 version CaRMetal permits some turtle graphics (programmed in JavaScript) either in 2D or in 3D.
The first working Logo turtle robot was created in 1969. A display turtle preceded the physical floor turtle. Modern Logo has not changed very much from the basic concepts predating the first turtle. The first turtle was a tethered floor roamer, not radio-controlled or wireless. At BBN Paul Wexelblat developed a turtle named Irving that had ...
A Cartesian coordinate surface in this space is a coordinate plane; for example z = 0 defines the x-y plane. In the same space, the coordinate surface r = 1 in spherical coordinates is the surface of a unit sphere, which is curved. The formalism of curvilinear coordinates provides a unified and general description of the standard coordinate ...
This product is angle addition since x = cos(A) and y = sin(A), where A is the angle that the vector (x, y) makes with the vector (1,0), measured counter-clockwise. So with ( x , y ) and ( t , u ) forming angles A and B with (1, 0) respectively, their product ( xt − uy , xu + yt ) is just the rational point on the unit circle forming the ...
Let the points on the circle be a sequence of coordinates of the vector to the point (in the usual basis). Points are numbered according to the order in which drawn, with n = 1 {\displaystyle n=1} assigned to the first point ( r , 0 ) {\displaystyle (r,0)} .
A unit -ball is a line segment whose points have a single coordinate in the interval [,] of length , and the -sphere consists of its two end-points, with coordinate {,} .