A New Geometry of i
Part Two: Complex Slope in 3Di Coordinates
by Greg Ehmka
Abstract
This article is part two of a series. Part one may be reviewed here. 3Di coordinates allows for a clear intuitive geometric interpretation of complex slope which is the sum of real slope and imaginary slope. Real slope occurs in a “front” view and can be visualized as the climb or descent of an aircraft. Imaginary slope occurs in the “top” view and can be visualized as the aircraft’s heading. When the two occur together they generate complex slope. The algebra is a straightforward extension of the usual slope-intercept equation to include complex numbers. An ‘inverse imaginary slope’ may also be defined. This can be visualized as standing at the edge of a runway with the aircraft taking off going away from you. It appears to rise vertically. These three different two dimensional slopes transform into one another as the lines are rotated.
Contents
Complex Slope
An Intuitive Model
Real, Imaginary and Complex Slope
Example
Inverse Imaginary Slope
Table of Slopes in 3Di
Transformation of Two Dimensional Slope
Summary
Further Reading
Complex Slope
3Di coordinates allow for a relatively simple extension of the standard concept of slope. The standard concept of slope would, in this model, be termed ‘real only slope’ and this appears in the front view. To this is added ‘imaginary only’ slope which appears in the top view. Summing the two together generates ‘complex slope.’
An Intuitive Model
Using the animation above, visualize an aircraft taxiing down the runway prior to take-off. Our view is off to the side, with the taxiing aircraft moving from left to right. And, let’s say that exactly to the right is a heading of zero. Exactly in front of us, the aircraft reaches take-off speed and rotates to begin its climb. This is the violet ball at the origin. The violet line is the aircraft’s climb while maintaining the same heading. This is real slope and zero imaginary slope, sometimes referred to as ‘rise over run.’
Next, at the black ball, the aircraft reaches cruising altitude and levels off while maintaining the same heading. And, the black line shows its flight path with zero real slope and zero imaginary slope.
Next, at the red ball, the aircraft executes a 45-degree turn to the left while maintaining altitude. The red line shows its flight path with imaginary slope and zero real slope. This could be referred to as ‘glide over run.’
And finally, while maintaining that heading, at the blue ball, it begins another climb. The blue line then shows both real slope, which is the climb, and imaginary slope, which is the heading other than zero. So, in flight path terms, complex slope is the sum of climb/descent plus heading.
Real, Imaginary and Complex Slope
Removing the idea of an aircraft, since it has a direction and motion, and just focusing on the line segments, real only slope of a line in three dimensions is:
while imaginary only slope is:
Real only slope appears in the front view (FRP) and imaginary only slope appears in the top view (TIP).
Real only slope has the usual slope-intercept equation of a line:
and imaginary only slope would then have a corresponding slope-intercept equation of a line:
In graphing terms, real only slope is rise over run and imaginary slope would be glide over run. Either or both can be positive, negative or zero. Complex slope combines the two and is ‘rise plus glide over run’. The two equations can be combined to give:
Algebraically, complex slope extends standard slope by adding in the imaginary number for the glide. Since there are two slopes:
And the calculation of complex slope becomes:
Also, rather than an axis intercept, there is a displacement of the line relative to both the y-axis and the iz-axis. Meaning there is a real displacement and an imaginary displacement. So the complete equation is:
The real displacement moves the line up and down. The imaginary displacement moves the line forward and backward.
Example:
What is the equation of the line that goes through the two points: (3,2,i) and (1,-3,6i) ?
The first step is to calculate the two slopes, real and imaginary:
The second step is to insert the slopes along with either point into the basic equation to solve for the displacements. Using the first point:
The third step, if needed, is to insert the slopes and the second point into the basic equation to verify that the two points give the same displacements.
And so, the completed equation for the line with the two specified points is:
When this line is projected to the front view (real only slope) it appears as:
When the line is projected to the top view (imaginary only slope) it appears as:
The 3Di graph of the line along with the two specified points is as follows:
Inverse Imaginary Slope
In addition to real slope in the FRP and imaginary slope in the TIP, denoted by:
we can define an ‘inverse imaginary slope’:
and denote it by:
Real slope shows up in the front view and imaginary slope shows up in the top view. Inverse imaginary slope shows up in the right side view.
One way that this can be visualized is by standing at the end of a runway while the aircraft takes off going away from us. In this front view the aircraft appears to rise vertically. This vertical ascent appearance occurs in both front and top views. And, this demonstrates that a line with inverse imaginary slope, so defined, appears as a vertical line in both the FRP and the TIP.
Intuitively, we can carry these visualizations further to formally observe that:
(1) A line with real only slope shows up as a vertical line in the side view, and a line with zero slope in the top view.
(2) A line with imaginary only slope shows up as a line with zero slope in front view, and a line with zero slope in side view.
(3) And, as stated above, a line with inverse imaginary only slope shows up as a vertical line in both front and top views.
Inverse imaginary slope may appear somewhat counter intuitive in that the glide path of the above mentioned aircraft would have a positive inverse imaginary slope on landing/approach, and a negative inverse imaginary slope on take-off/departure.
Continuing the example with the two previously specified points, (3,2,i) and (1,-3,6i), the inverse imaginary slope can be calculated as:
This can be viewed, in the above last animation, as the blue line comes around to show the RSIP view; and it can be verified by projecting the line (blue) to the RSIP as follows. In the side view the axes are:
and the two displacements, when combined, project to a y-intercept that is different. By inserting the two points into the equation:
the y-intercept is calculated as:
So the equation of this line is:
And when this line is projected to the right side view it appears as:
The three different two dimensional graphs are generated by the following equations:
Table of Slopes in 3Di
Additionally, the three 2-dimensional slopes can also be visualized as the three planar rotations of a spacecraft. I.e., pitch, yaw and roll which is indicated in the fourth column of the table:
slope |
notation |
plane |
slope rotation |
action |
2D relationships |
complex | pitch + yaw | rise + glide over run | Slope in all three, FRP, TIP, RSIP | ||
real only | FRP | pitch | rise over run | Horizontal in TIP, vertical in RSIP | |
imaginary only | TIP | yaw | glide over run | Horizontal in both FRP and RSIP | |
inverse imaginary | RSIP | roll | rise over glide | Vertical line in both FRP and TIP |
Transformation of Two Dimensional Slope
If we specify a point at (1, 1/2 , 0i):
and then draw a line through this point with no displacement, meaning a line through this point and the origin:
The equation of this line is generated by:
With no displacement, zero imaginary slope and an arbitrary 1/2 real slope, the equation reduces to:
which is a line with real-only slope located on the front, real plane (FRP).
If we were to rotate this line about the x-axis using a ‘rotator coefficient’ i^{a} (see sections: 3.25, 3.7, 9.38 on this rotator in the eBook), the effect is to alter the line’s two dimensional real and imaginary slopes. Meaning:
So as a moves through its interval, the line is rotated about the x-axis:
line rotation about the x-axis
As the line rotates, its projected two dimensional slope transforms from real-only to complex to imaginary only to complex. And then to negative real-only to complex to negative imaginary-only to complex and then back again to positive real-only. So if we look at various values of a:
For a = 0 the slope is positive real only:
For each value of a; First is the 3Di view and second is the two dimensional view. The black line is the two dimensional real slope in the front view and the red line is the two dimensional imaginary slope in the top view:
For a = .6 the slope is complex:
For a = 1 the slope is positive imaginary only:
For a = 1.6 the slope is complex with negative real and positive imaginary:
For a = 2.6 the slope is complex with negative real and negative imaginary:
For a = 3 the slope is negative imaginary only:
As stated before, if a displacement is added to the line, a real displacement moves the line up and down and an imaginary displacement moves the line forward or backward. Then the line will rotate at the displacement point about a line (green line below) through that point and parallel to the x-axis. For example, adding a displacement and using the above line with a = 3:
Just as this line, which is rotated about the x-axis, transitions between real and imaginary slope, if the line is rotated about the iz-axis the slope will transition between real and inverse imaginary slopes. Similarly if the line is rotated about the y-axis the line will transition between imaginary and inverse imaginary slopes.
Summary
1) An intuitively clear geometric interpretation of complex slope is provided by the modeling of an aircraft. Real slope is the aircraft’s climb or descent. Imaginary slope is the aircraft’s heading. Complex slope is the sum of the two.
2) A line occurring in three dimensions may be projected to any of three two dimensional planes. The front, real plane, the top imaginary plane and the right side imaginary plane. Real slope occurs in the front view. Imaginary slope occurs in the top view.
3) A third type of two dimensional slope, that being, ‘inverse imaginary slope’ can be defined and occurs in the right side view. This is visualized by standing at the end of a runway with an aircraft taking off going away from you. In two dimensions it appears to rise vertically.
4) Complex slope along with the three two-dimensional slopes can be summarized in the following table:
slope |
notation |
plane |
slope rotation |
action |
2D relationships |
complex | pitch + yaw | rise + glide over run | Slope in all three, FRP, TIP, RSIP | ||
real only | FRP | pitch | rise over run | Horizontal in TIP, vertical in RSIP | |
imaginary only | TIP | yaw | glide over run | Horizontal in both FRP and RSIP | |
inverse imaginary | RSIP | roll | rise over glide | Vertical line in both FRP and TIP |
5) Any given line may be rotated about any of the three axes. The complex slope and the three two-dimensional slopes change accordingly. As such, a rotating line will transform its two dimensional slope into one of the other two dimensional slopes and back again.. For example; A line rotating about the x-axis will transform from real only to complex to imaginary only, back to real and so on. There are various possibilities depending on how the line is initially defined and about which axis it is rotating.
Further Reading
1) The third part in this series will likely be entitled: “Closed Surface Functions in 3Di Coordinates”. Briefly, if we define a ‘rectangular complex domain’ with intervals, for example;
as input to a function, various closed surfaces e.g. the sphere, cube, lozenges, pointed cylinders, barrels and so on result from a four dimensional function of the form:
Since any closed surface will have at least two points on any vertical line, except for tangents, it is the fourth variable, iv, that allows for the relationships to be actual functions passing the vertical line test. These are one to one functions with a single complex number input and a single complex number output. As such, inverse functions may also be generated.
2) See the eBook: “A Three Dimensional Coordinate System for Complex Numbers” for the original material and a wide array of other topics including historical curves, transcendental functions, open surfaces and 4Di, 4Dii and 6Dii animations in time.
3) A gallery of interesting surface graphs from the eBook, with some of their equations, is available on Facebook.