Two aircraft in flight define a point in 8-dimensional state space. The collision condition carves out a hypersurface. Points inside it are dangerous. Parallel coordinates make this visible.
An aircraft in 2D airspace has four numbers — position and velocity. Two aircraft have eight, and the pair's state is one point in 8D. "Will they collide?" is a question about where that point falls relative to a surface.
Collision detection was Inselberg's original motivation for parallel coordinates. The Cartesian test computes a high-dimensional distance; in parallel coordinates it becomes a visual pattern.
Consider two aircraft, A and B, each with a position and velocity in two dimensions. Aircraft A has state and aircraft B has state . The combined state is an 8-dimensional vector.
At any future time , the positions are:
The distance between them at time is:
This is a quadratic in . Its minimum, the closest point of approach (CPA), occurs at:
where and . The CPA distance is . When this value drops below a safety threshold , the pair is on a collision course.
Drag either aircraft's position (filled circle) or velocity arrow tip (open circle) in the overhead view.
Figure 1. Top: overhead map of two aircraft with velocity vectors. Bottom: parallel coordinates with 8 axes representing the joint state. The display turns red when the closest point of approach falls below the safety threshold.
Most random trajectory pairs are safe; the dangerous ones cluster. The condition defines a region in 8D space. In the reduced CPA-time vs. CPA-distance view, dangerous trajectories sit in a band near the bottom.
The boundary of the dangerous region is a hypersurface. Its envelope representation is a structured pattern — that structure is what makes visual detection possible.
Brush vertically in the scatter plot to select dangerous approaches. The corresponding polylines highlight in the parallel coordinates view.
Figure 2. Left: CPA time vs. CPA distance for 100 random trajectory pairs. Points below the dashed threshold line are dangerous. Right: the 8D states of all pairs in parallel coordinates. Brush the dangerous region to see what their polylines look like.
A human controller can't evaluate the CPA formula for every pair, but the clustering of dangerous polylines means a trained eye can pick out hazardous configurations visually.
A single pair of aircraft traces a path through the 4D position space as time advances. At each moment, their positions define a polyline in parallel coordinates. Over 60 time steps, these polylines sweep through the display. When the aircraft converge, the polylines converge toward a common region. When they diverge, the polylines fan out.
The convergence of the trajectory is visible as the darkening polylines funnel together, then separate after closest approach.
Figure 3. One pair of aircraft over 60 time steps, shown as polylines in the 4D position space (x_A, y_A, x_B, y_B). Color darkens with time. Convergence of trajectories produces convergence of polylines toward a common region.
The safe/dangerous boundary is a hypersurface in 8D. In parallel coordinates it has an envelope — a bounding curve separating safe from dangerous polylines.
This is the containment test from chapter 6: a trajectory pair is safe iff its polyline lies outside the dangerous envelope. The geometry doesn't care whether the surface encloses a polytope or a collision region.
Figure 4. The separation envelope in parallel coordinates. Safe polylines (green) stay outside the shaded dangerous region. Dangerous polylines (red) pass through it. Adjust the threshold to widen or narrow the boundary. The envelope's shape is the hypersurface's PC signature.
Inselberg started from this problem — detecting dangerous proximity in high-dimensional state spaces — and worked backward to the geometry. The hypersurface representation, envelope construction, and containment test emerged from the need to see collision conditions that can't be drawn in Cartesian space.