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For any point, the abscissa is the first value (x coordinate), and the ordinate is the second value (y coordinate). In mathematics, the abscissa (/ æ b ˈ s ɪ s. ə /; plural abscissae or abscissas) and the ordinate are respectively the first and second coordinate of a point in a Cartesian coordinate system: [1] [2]
Likewise, (x, −y) are the coordinates of its reflection across the first coordinate axis (the x-axis). In more generality, reflection across a line through the origin making an angle with the x-axis, is equivalent to replacing every point with coordinates (x, y) by the point with coordinates (x′,y′), where
This can be seen by holding your hands together with palms up and fingers curled. If the curl of the fingers represents a movement from the first or x-axis to the second or y-axis, then the third or z-axis can point along either right thumb or left thumb.
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.
A point is chosen as the pole and a ray from this point is taken as the polar axis. For a given angle θ, there is a single line through the pole whose angle with the polar axis is θ (measured counterclockwise from the axis to the line). Then there is a unique point on this line whose signed distance from the origin is r for given number r.
The only difference is that Tait–Bryan angles represent rotations about three distinct axes (e.g. x-y-z, or x-y′-z″), while proper Euler angles use the same axis for both the first and third elemental rotations (e.g., z-x-z, or z-x′-z″). This implies a different definition for the line of nodes in the geometrical construction.
To be able to plot points, one needs to first decide on a point in plane which will be called the origin, and a couple of perpendicular lines, called the x and y axes, as well as a preferred direction on each of the lines. Usually one chooses the x axis pointing right and the y axis pointing up, and these will be named the positive directions ...
Selecting reference points. In two dimensions, given an ordered set of three or more connected vertices (points) (such as in connect-the-dots) which forms a simple polygon, the orientation of the resulting polygon is directly related to the sign of the angle at any vertex of the convex hull of the polygon, for example, of the angle ABC in the picture.