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For positive y- and z-axis, we have to face two different conventions: In case of land vehicles like cars, tanks etc., which use the ENU-system (East-North-Up) as external reference (World frame), the vehicle's (body's) positive y- or pitch axis always points to its left, and the positive z- or yaw axis always points up. World frame's origin is ...
When viewed at a position along the positive z-axis, the ¼ turn from the positive x-to the positive y-axis is counter-clockwise. For left-handed coordinates, the above description of the axes is the same, except using the left hand; and the ¼ turn is clockwise. Interchanging the labels of any two axes reverses the handedness.
The axes may then be referred to as the x-axis, y-axis, and z-axis, respectively. Then the coordinate planes can be referred to as the xy-plane, yz-plane, and xz-plane. In mathematics, physics, and engineering contexts, the first two axes are often defined or depicted as horizontal, with the third axis pointing up.
The Z axis is the line between the North and South Poles, with positive values increasing northward. In WGS 84, this is the international reference pole (IRP), which does not exactly coincide with the Earth's rotational axis [6] The slight "wobbling" of the rotational axis is known as polar motion, and can actually be measured against an ECEF. [7]
The user may choose to replace the inclination angle by its complement, the elevation angle (or altitude angle), measured upward between the reference plane and the radial line—i.e., from the reference plane upward (towards to the positive z-axis) to the radial line. The depression angle is the negative of the elevation angle.
Suppose a rectangular xyz-coordinate system is rotated around its z axis counterclockwise (looking down the positive z axis) through an angle , that is, the positive x axis is rotated immediately into the positive y axis. The z coordinate of each point is unchanged and the x and y coordinates
In the cylindrical coordinate system, a z-coordinate with the same meaning as in Cartesian coordinates is added to the r and θ polar coordinates giving a triple (r, θ, z). [8] Spherical coordinates take this a step further by converting the pair of cylindrical coordinates (r, z) to polar coordinates (ρ, φ) giving a triple (ρ, θ, φ). [9]
If the determinant is negative, then the polygon is oriented clockwise. If the determinant is positive, the polygon is oriented counterclockwise. The determinant is non-zero if points A, B, and C are non-collinear. In the above example, with points ordered A, B, C, etc., the determinant is negative, and therefore the polygon is clockwise.