Calculating the Angles of Depression

 

The angle of depression refers to the angle between an object's horizontal axis and its vertical axis. An object's axis of rotation is always in a state of dynamic tension, referred to as angular momentum. The more the angle of depression with respect to the horizontal axis, the higher the object's velocity. Therefore objects which have high angles of depression are going to fly away from you faster than objects with low angles of depression. The figure below depicts this phenomenon in detail.

Let us start with the equation. The angle of depression can be written as the slope of the tangent (which is the horizontal axis) to the viewing angle. If we plot this angle against the horizontal axis, we get the tangential component of the angle of position. The slope of the tangent is equal to the cosine of the angle of position. Therefore, when we determine the angle of tilt against the horizontal axis, we find that it is equal to -pi/n where n is the focal length of the optical system and a is the angle of the bore on the vertical axis. Therefore, if the angle of depression were zero then we would simply observe any point on the horizon.

We now proceed to the third rule of the depression formula, the arctangent. The angle of depression, which is again expressed as the angle of the tangent, between the observer and an object will be calculated as the arctangent of the triangles formed by their hypotenuse. For an object whose distance from the observer is equivalent to the hypotenuse, this expression yields a negative number, corresponding to zero degrees of angle of position. Conversely, for an object whose distance from the observer is equal to the hypotenuse, then the angle of depression formula yields a positive number, corresponding to the exact magnitude of the angle of tilt against the horizontal axis.

A similar argument can be used to calculate the angle of the tangent on the x-axis. To do this, plot the tangent of the y-axis on a graph, with the x-axis positioned near the observer. Then calculate the angle of curvature against the x-axis along the x-axis in degrees. If t is the time for the arrival at that location on the earth, then the tangent must be halfway between zero and half of the observed time, which is about 0 degrees.

If we plot the tangent on the y-axis, then it must be halfway between zero and one hundred sixty degrees. From the above calculations it should be obvious that the tangent of an object on the x-axis, when graphed, must always be expressed as a positive number. This gives rise to the fourth rule, namely, that the true angle of an ellipse is half of the angle of rotation. For any two points on the equator, the resulting angle on the map is half of the angle of the corresponding point on the x-axis if the coordinates are plotted on a map between east and west, north and south, or west to north and east, or north to south.

To find these angles, note that the observer has sight east, northwest, northeast, and southeast. The angles of the compass directions can be calculated from the angles described by the coordinate system. The formula for finding the angles is similar to the formula used in Physics for determining acceleration. The observer chooses a point on the equator, whose distance from the center of the earth is equal to the radius of the earth. Then the angle of projection of that point onto the x-axis is given by the formula given below.

Let's consider an example. First note that the observer determines the horizontal angle by measuring east, west, and north with respect to the horizon. Then, for the sake of simplicity, we assume that the observer chooses points A, B, and C at equal distances from the observer on the x-axis. Then he maps (at right angles to the x-axis) the distance from point A to point B on the x-axis to the angle of the inner circle formed by the inner lines of the circle of the equal horizontal parallels, also known as the parallels vertically, along the equator to the angle of the outer circle of the equator to the angle of the circle's outer line. This becomes (B, C, D, E, F) the inner circle formed by the parallel lines. This formula can be applied to any equatorial plane.

Now, let's use the same formula for each point on the equator. We calculate the angle of symmetry from each point by multiplying the formula in place of the left triangle with the corresponding angle on the right triangle. The formula then gives us the right triangle where the equator and the circle meet. This gives the right angle of recession.