With any given 3Di space function, , there will be six resulting two-dimensional ‘projection planes’ which are the two sides of the three colored planes shown below.
Of the six resulting two-dimensional planes, two of the planes are ‘real planes’ and four of the planes are ‘imaginary planes’, as follows:
i.Both sides of the violet plane – horizontal axis (real) and vertical axis (real) – , which we will call the front and rear real planes.
ii.Both sides of the red plane – horizontal axis (real) and depth axis (imaginary) – , which we will call the top and bottom imaginary planes.
iii.Both sides of the blue plane – depth axis (imaginary) and vertical axis (real) – which we will call the left and right side imaginary planes.
In practice we will rarely, if ever, use the rear and bottom planes and only occasionally use the left side plane.
In the coordinate system ‘3Di’, the normal conic hyperbola and normal conic ellipse are two 2D views (orthogonal to one another) of the same 3D object!
To illustrate the basic idea here is the usual equation of a hyperbola:
And, with a = b = 1 and C = 1,
And, the normal graph in 2D:
If we take another look at the graph, and this time ALSO incude the interval
The graph line joining the vertices actually graphs the real value of y, which is zero at each point, if the domain of x is allowed to take on values between 1 and -1.
With C , in a normal 2D graph, there is a gap between the two vertices in the x, y plane (FRP, x is horizontal, y is vertical). This is because the values for x between the two vertices, if inserted into the equation, produce complex numbers and are therefore not usually shown. Now that we have an interpretation for these numbers, and allow them as part of the domain, what shows up in three dimensions is a circle in the plane (TIP, as horizontal, and as vertical) in between the vertices!
The associated ellipse/circle in TIP:
And then, seeing the three dimensional view:
Animation 3 ’3Di Conic’
The TIP circle is orthogonal to the FRP hyperbola, and is not visible in the Front Real Plane (FRP) view, because the normal 2D plane graphs are projections. The circle only becomes visible in a 3D view, or when viewing the Top Imaginary Plane (TIP) directly, that is, in top view.
See: Section 4.1 Quadratic Input and Output for further discussion of the vertices and ‘bifurcation.’
Additionally, notice that when square roots are taken, two three-dimensional functions are generated: one for each root. See next section.