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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 = =.
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 ...
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 ...
Gauss's circle problem asks how many points there are inside this circle of the form (,) where and are both integers. Since the equation of this circle is given in Cartesian coordinates by x 2 + y 2 = r 2 {\displaystyle x^{2}+y^{2}=r^{2}} , the question is equivalently asking how many pairs of integers m and n there are such that
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.
A point P has coordinates (x, y) with respect to the original system and coordinates (x′, y′) with respect to the new system. [1] In the new coordinate system, the point P will appear to have been rotated in the opposite direction, that is, clockwise through the angle . A rotation of axes in more than two dimensions is defined similarly.
This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reverse the definitions of θ and φ): . The polar angle is denoted by [,]: it is the angle between the z-axis and the radial vector connecting the origin to the point in question.