Huzita's axioms
Huzita's axioms are a set of rules related to the mathematical principles of origami. They were formulated by ItalianJapanese mathematician Humiaki Huzita in 1992, and are the most powerful known set of axioms related to origami. The axioms are as follows:
 Given two points p_{1} and p_{2}, there is a unique fold that passes through both of them.
 Given two points p_{1} and p_{2}, there is a unique fold that places p_{1} onto p_{2}.
 Given two lines l_{1} and l_{2}, there is a unique fold that places l_{1} onto l_{2}.
 Given a point p_{1} and a line l_{1}, there is a unique fold perpendicular to l_{1} that passes through point p_{1}.
 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}.
 Given two points p_{1} and p_{2} and two lines l_{1} and l_{2}, there is a fold that places p_{1} onto l_{1} and p_{2} onto l_{2}.
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Details
Axiom 1
Given two points p_{1} and p_{2}, there is a unique fold that passes through both of them.
In parametric form, the equation for the line that passes through the two points is:
Axiom 2
Given two points p_{1} and p_{2}, there is a unique fold that places p_{1} onto p_{2}.
This is equivalent to finding the perpendicular bisector of the line segment p_{1}p_{2}. This can be done in four steps:
 Use Axiom 1 to find the line through p_{1} and p_{2}, given by
 Find the midpoint of p_{mid} of P(s'')
 Find the vector v^{perp} perpendicular to P(s)
 The parametric equation of the fold is then:
Axiom 3
Given two lines l_{1} and l_{2}, there is a unique fold that places l_{1} onto l_{2}.
This is equivalent to finding the bisector of the angle between l_{1} and l_{2}. Let p_{1} and p_{2} be any two points on l_{1}, and let q_{1} and q_{2} be any two points on l_{2}. Also, let u and v be the unit direction vectors of l_{1} and l_{2}, respectively; that is:
Axiom 4
Given a point p_{1} and a line l_{1}, there is a unique fold perpendicular to l_{1} that passes through point p_{1}.
This is equivalent to finding a perpendicular to l_{1} that passes through p_{1}. If we find some vector v that is perpendicular to the line l_{1}, then the parametric equation of the fold is:
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:
Axiom 6
Given two points p_{1} and p_{2} and two lines l_{1} and l_{2}, there is a fold that places p_{1} onto l_{1} and p_{2} onto l_{2}.This axiom is equivalent to finding a line simultaneously tangent to two parabolas, and can be considered equivalent to solving a thirddegree equation. The two parabolas have foci at p_{1} and p_{2}, respectively, with directrices defined by l_{1} and l_{2}, respectively.