That way you can specify all points on an infinite plane. Descartes' coordinate system created a link between algebra and geometry. Geometric shapes, such as circles, could now be described algebraically using the coordinates of the points that make up the shapes. Using such a coordinate system it is possible to solve geometric problems using algebra, and vice versa.
Unfortunately, Fermat never published his treatise, although he shared his ideas with other mathematicians such as Blaise Pascal — In the eyes of the scientific world, the publication date of a technical paper determines when a new idea or invention is released into the public domain. Consequently, ever since this publication Descartes has been associated with the xy-plane, which is why it is called the Cartesian plane. If Fermat had been more efficient in publishing his research results, the xy-plane would have been called the Fermatian plane!
Boyer and Merzbach, In the same way one defines a Cartesian space of any dimension n , whose points can be identified with the tuples lists of n real numbers, that is, with. The Euclidean distance between two points of the plane with Cartesian coordinates and is. This is the Cartesian version of Pythagoras' theorem.
In three-dimensional space, the distance between points and is. Translating a set of points of the plane, preserving the distances and directions between them, is equivalent to adding a fixed pair or numbers X , Y to the Cartesian coordinates of every point in the set.
That is, if the original coordinates of a point are x , y , after the translation they will be:. To make a figure larger or smaller is equivalent to multiplying the Cartesian coordinates of every point by the same positive number m.
If x , y are the coordinates of a point on the original figure, the corresponding point on the scaled figure has coordinates. If m is greater than 1, the figure becomes larger; if m is between 0 and 1, it becomes smaller.
To rotate a figure counterclockwise around the origin by some angle is equivalent to replacing every point with coordinates x , y by the point with coordinates x ' , y ' , where:. The Euclidean transformations of the plane are the translations, rotations, scalings, reflections, and arbitrary compositions thereof. The result of applying a Euclidean transformation to a point is given by the formula:.
This is equivalent to saying that A times its transpose must be a diagonal matrix. If these conditions do not hold, the formula describes a more general affine transformation of the plane. The formulas define a translation if and only if A is the identity matrix. The transformation is a rotation around some point if and only if A is a rotation matrix , meaning that. Fixing or choosing the x -axis determines the y -axis up to direction.
Namely, the y -axis is necessarily the perpendicular to the x -axis through the point marked 0 on the x -axis. But there is a choice of which of the two half lines on the perpendicular to designate as positive and which as negative. Each of these two choices determines a different orientation also called handedness of the Cartesian plane. The usual way of orienting the axes, with the positive x -axis pointing right and the positive y -axis pointing up and the x -axis being the "first" and the y -axis the "second" axis is considered the positive or standard orientation, also called the right-handed orientation.
A commonly used mnemonic for defining the positive orientation is the right hand rule. Placing a somewhat closed right hand on the plane with the thumb pointing up, the fingers point from the x -axis to the y -axis, in a positively oriented coordinate system. The other way of orienting the axes is following the left hand rule , placing the left hand on the plane with the thumb pointing up. When pointing the thumb away from the origin along an axis, the curvature of the fingers indicates a positive rotation along that axis.
Regardless of the rule used to orient the axes, rotating the coordinate system will preserve the orientation. Switching any two axes will reverse the orientation. Once the X - and Y -axes are specified, they determine the line along which the Z -axis should lie, but there are two possible directions on this line. The two possible coordinate systems which result are called 'right-handed' and 'left-handed'.
The standard orientation, where the XY -plane is horizontal and the Z -axis points up and the X - and the Y -axis form a positively oriented two-dimensional coordinate system in the XY -plane if observed from above the XY -plane is called right-handed or positive. The name derives from the right-hand rule. If the index finger of the right hand is pointed forward, the middle finger bent inward at a right angle to it, and the thumb placed at a right angle to both, the three fingers indicate the relative directions of the X -, Y -, and Z -axes in a right-handed system.
The thumb indicates the X -axis, the index finger the Y -axis and the middle finger the Z -axis. This website uses cookies to improve your experience. We'll assume you're ok with this, but you can opt-out if you wish. Read More. Table of Contents. How much is a silver quarter dollar worth? Can you cancel a rental agreement? Accept Decline Cookie Settings. I consent to the use of following cookies:.
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