General Equation of a Curve
Summary: The single equation in two coordinates obtained by substituting the full rigid motion , (with ) into a curve’s equation. It contains four free parameters and therefore subsumes every equation of the curve in any orthogonal coordinate system. Equivalence of two equations — i.e., whether they represent the same curve — reduces to solving for these parameters.
Sources: chapter2
Last updated: 2026-04-24
Definition (§35)
Given a curve and any one equation for it in orthogonal coordinates , the substitution
produces an equation in that holds in any rotated-and-translated orthogonal coordinate system (see coordinate-transformations). Since no axis “lying in the same plane with the curve can be conceived which is not included in this last determination” (source: chapter2, §35), every possible orthogonal equation of appears as a specialization. For this reason Euler calls the resulting equation in the general equation of the curve.
The general equation is not a new equation in the usual sense — it is the original equation with four auxiliary symbols held formal, to be specialized by choice.
Deciding whether two equations represent the same curve (§36)
Suppose we are handed two equations — one in , one in — and we want to know if they define the same curve. The recipe:
- In the first equation substitute and (with ).
- Ask whether the resulting equation (now in , with formal parameters ) is proportional to the second equation for some choice of those parameters.
- If yes, the two equations describe the same curve; if no, they describe different curves.
This is Euler’s answer to the problem he flagged at §24. It is decisive but “rather tedious” (§38), so for equations of high degree he relies on shortcuts — in particular degree invariance; see degree-invariance.
Worked example (§36)
Claim: and describe the same curve, even though they look unrelated.
Substitute , into :
Scale to match the leading quadratic terms. Multiply the above by and the target equation by , so both have a leading term:
Match the quadratic coefficients. From : , i.e. . From : , consistent with the same ratio. With :
Match the linear coefficients. The -coefficient gives , and the -coefficient gives . Solving each for ,
The two values agree, so the linear terms match.
Match the constant. The transformed equation has constant , the target has constant . This gives . Since was not previously determined, no conflict arises.
With , , , the two equations agree up to the overall factor (absorbed by the -vs- scaling), so they define the same curve (source: chapter2, §36).
Geometrically: the target equation is the semicubical/parabolic curve (a parabola with axis along the -direction) viewed from an orthogonal coordinate system rotated by angle and translated by .
When equations must be different
If two equations pass the degree check (see degree-invariance) they may or may not be equivalent — one must run the full substitution to decide. But if they fail the degree check, they cannot be equivalent. This is the cheap half of the decision problem and is how Euler usually proceeds in practice.