You are may be thinking about Projective geometry and parallel lines here. So - that parallel lines meet at infinity - not perpendicular lines, as they meet at their point of intersection in the plane. So, I’m going to assume you mean “parallel lines” in my answer. You may know that when an artist draws a railway line going off to infinity they add a “point at infinity”, or “vanishing point”

Vanishing point, eastern Montana by Roy Luck, on Flickr

So - this theory adds a “line at infinity” to two dimensional Euclidean geometery, and then parallel lines are said to intersect on that line. The result is mathematically elegant because then any two points determine a line and any two lines determine a point.

Indeed there’s an interesting symmetry in the axioms and proofs of projective geometry - what mathematicians call a “duality”. Any theorem about points and lines remains true if you swap points and lines all the way through it. So it’s a neat theory with elegant proofs, which turns into Euclidean geometry when you remove that hypothetical line at infinity, It leads to surprising connections between mathematical proofs in geometry. This shows how the duality works:

Here a “complete quadrangle” is a set of four points with no three on the same line, and all the lines that pass through them. A “complete quadrilateral” is a sot of four lines, no three of which pass through the same point, together with all the points resulting from their intersections.

When you change lines to points and vice versa, then the definition of a quadrangle turns into a definition of a quadrilateral. More generally any theorem in projective geometery about quadrangles will also be true about quadrilaterals if you change points to lines and vice versa all the way through the proof. And more generally any definition, proof or theorem remains valid when you interchange lines with points and vice versa.

In this figure, each red point on the left turns into a red line on the right and each blue line turns into a red point when you do the duality. So, for example, a result about four points determining lines (as in the left hand diagram) becomes a result about four points determining lines (as in the right hand diagram).

You can visualize this line at infinity as a kind of circle at infinity - but there is one difference from the artist’s line or plane at infinity. It’s a strange circle because an artist would normally think of a railway line as having two “vanishing points because it vanishes to infinity in both directions. Normally you would only photograph it in one direction at a time but you’d think the opposite directions as two distinct points.

However to get the duality of projective geometery to work, we need two parallel lines intersect at only one point. To make that work, we need to identify opposite points as being the “same point”.

So it’s really a half circle, with the second half of the circle being treated as the first half traced again. But you call it a straight line in projective geometry. When you do that the result is a very elegant mathematical theory.

There’s another way to think of it is as the lines through the origin in 3D - which you treat as the points.

Here is an introduction to projective geometry.

This is the wikipedia article, but many of their articles on geometry are rather technical to the extent it’s often hard to know what they are saying unless you are a mathematician already somewhat familiar with the subject: Line at infinity.

You can do the same thing in three dimensions, this time the geometry has points, lines and planes. It relates to artistic drawings in full 3D with vanishing points in all directions. It’s related to the informal geometry that artists work with when they draw shapes with parallel lines in any direction, not just horizontal lines, with multiple vanishing points not just around the horizon but in any direction, above or below as well as around the horizon.

This time, to describe the geometry in an axiom system like Euclidean geometry, you’d start with 3D Euclidean geometry with its lines, planes and points, but to complete it you now need to add a Plane at infinity. That then works the same way as the line at infinity that you have to identify opposite points to make the duality work. For instance if you use a vanishing point at infinity vertically upwards, then you identify it as the same point as the vanishing point vertically downwards.

3D projective geometry can get a bit mind boggling even for those used to 2D projective geometry. You can think of as being like a sphere at infinity with opposite points identified, but in 3D projective geometry it’s called a plane. This time it’s not points and lines that are dual, rather, points and planes . Any two planes determine a line, and any two points determine a line - so two parallel planes will determine a line at infinity, just as in 2D projective geometry two parallel lines determine a point at infinity. Two of those lines at infinity will intersect at a point at infinity too, so you need not just a line and points at infinity - you need a plane at infinity with lines and points in it.

It’s not used as much as 2D projective geometry, but it’s elegant too.

See also:

Plane Projective Geometry (online book)