. They were formulated by
related to origami. The axioms are as follows:
It should be noted that Axiom (5) may have 0, 1, or 2 solutions, while Axiom (6) may have 0, 1, 2, or 3 solutions.
Axiom 5
Given two points p_{1} and p_{2} and a line l_{1}, there is a fold that places p_{1} onto l_{1} and passes through p_{2}.
This axiom is equivalent to finding the intersection of a line with a circle, so it may have 0, 1, or 2 solutions. The line is defined by l_{1}, and the circle has its center at p_{2}, and a radius equal to the distance from p_{2} to p_{1}. If the line does not intersect the circle, there are no solutions. If the line is tangent to the circle, there is one solution, and if the line intersects the circle in two places, there are two solutions.
If we know two points on the line, (x_{1}, y_{1}) and (x_{2}, y_{2}), then the line can be expressed parametrically as:

Let the circle be defined by its center at p_{2}=(x_{c}, y_{c}), with radius r equal to the distance from p_{1} to p_{2}. Then the circle can be expressed as:
In order to determine the points of intersection of the line with the circle, we substitute the x and y components of the equations for the line into the equation for the circle, giving:
Or, simplified:
Where:


Then we simply solve the quadratic equation:
If the discriminant b^{2}4ac < 0, there are no solutions. The circle does not intersect or touch the line. If the discriminant is equal to 0, then there is a single solution, where the line is tangent to the circle. And if the discriminant is greater than 0, there are two solutions, representing the two points of intersection. Let us call the solutions d_{1} and d_{2}, if they exist. We have 0, 1, or 2 line segments:

A fold F_{1}(s) perpendicular to l_{1} through its midpoint will place p_{1} on the line at location d_{1}. Similarly, a fold F_{2}(s) perpendicular to l_{2} through its midpoint will place p_{1} on the line at location d_{2}. The application of Axiom 2 easily accomplishes this. The parametric equations of the folds are thus:
