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## Path Curves in All Space

The all real case

The Line

When we step a point along a line, we have recourse to two directing points and an intermediate line. The result is a pair of conjugate, linear measures in the line, between two invariant end-points that divide the line (i.e., points that cannot be stepped, or, if stepped, step exactly to where they are) .

Progression through the elements-

linear measure

stepping a point along a line

The Plane

When we step a point across a plane, we have recourse to two stepping lines, and our stepping point in the plane is at their intersection. Our lines rotate in points, which are the vertices of a triangle, and are guided by linear measures in the sides of that triangle, and the result is a path curve in the plane. The sides of the triangle cannot be stepped, so are invariant. Accordingly, the entire triangle is invariant.

planar measure

stepping a point across a plane

All Space
Path Curves in all Space

When we want to step a point across space, we must have recourse to stepping planes, and the stepping point will be at their intersection.

Three planes intersect in a point, so we must have a minimum of three stepping planes.

three planes can track a point across space

Where for the planar case, we have lines rotated in points between lines, now for all space we must have stepping planes rotating in linesbetween planes, of which there will be one pair per line.

Now, a pair of planes have a line in common, so there will be two distinct lines-in-common for two, distinct pairs of planes:

If these two lines are skew to each other, then the four planes intersect in the six lines of a tetrahedron. (In fact, this is how four planes generally intersect.)

The vertices of it lie at the four, distinct places where three of the four planes intersect.

Opposite sides are skew, so there are three such skew pairs.

All of these elements, point, line or plane, are real, as opposed to imaginary .

Four general planes form a tetrahedron.

they intersect

• by triples in 4 points (vertices)
• by pairs in 6 lines (sides)

The four planes "enclose" volumes of space. That is, they divide the volume of all space into five volumes. If one of these is not cut by the plane at infinity, the remaining four must be.

We might take these four volumes cut by the plane at infinity as "external", and the fifth, uncut volume as "internal"; that is, as the "inside", of the tetrahedron. (Contrast this with the "enclosure" of a cone.)

"Outside" and "Inside"

We see that any one of these six could be a line in which a stepping plane might rotate, and that while six are available, three will do for tracking our point as it steps through the volume of space. Three such are shown one at a time on the right.

A transforming plane rotates between two invariant facets of the tetrahedron.

The rotations are guided by (independently determined) measures in the sides opposite to, and skew to, the sides in which the planes rotate.

The result is a path curve moving in space from one vertex of the tetrahedron to another, and avoiding the other two vertices.

In the construction on the left, the red path curve running through the “volume” of the tetrahedron presents as an intersection of two path cones.

The black path curve is a plane section of a path cone with its vertex at D.

The blue path curve is a plane section of a path cone with its vertex at A.

You may change the black path curve in tetrahedral face ABC by dragging points K and L, and the blue path curve in face BCD by dragging points E and F.

The red path curve adapts to these changes: it appears to “drift” between the blue and black curves.