Polynomial Curves

Polynomial curves are smooth, continuous graphs defined by equations involving powers of variables. Their elegance lies in the way algebraic expressions give rise to geometric forms — from simple lines to intricate higher-degree curves. In homogeneous coordinates, we write points as (x : y : z) instead of (x, y), which makes the curve scale-invariant and allows us to include points at infinity. This projective perspective is essential for understanding the full behavior of polynomial curves, especially in geometric constructions.

Within the Encyclopedia of Poly Geometry (EPG), we typically work with homogeneous barycentric coordinates, which are well-suited for triangle-based constructions and affine-to-projective transitions. Occasionally, we also use homogeneous projective coordinates.

A key property of polynomial curves is that they are uniquely determined by a finite number of points. For curves of degree n, the number of points required follows a quadratic pattern.

Cramer’s Theorem on algebraic curves

Theorem

A polynomial curve of degree n in the projective plane is uniquely determined (up to scale) by n(n + 3)/2 generic points.

Explanation

In homogeneous coordinates, a polynomial of degree n in three variables x, y, z has (n + 1)(n + 2)/2 monomial terms. Since projective curves are defined up to scale, we subtract one degree of freedom, resulting in n(n + 3)/2 independent coefficients. Each point imposes a linear constraint, so this number of generic points suffices to determine the curve uniquely.

This result, known as Cramer’s Theorem, was formulated by Swiss mathematician Gabriel Cramer in the 18th century. It underpins many constructive questions in the Encyclopedia of Poly Geometry, such as: How many points are needed to reconstruct a curve of degree n ? Which combinations of points and lines determine a unique curve?

Examples: Number of Points Required to Determine a Curve
Degree n	Curve Type	Points Required
1	Line	2
2	Conic	5
3	Cubic	9
4	Quartic	14

n	…	n(n + 3)/2

Bézout’s Theorem

Theorem

Two projective plane curves of degrees m and n, with no common component, intersect in exactly m · n points — counted with multiplicity and including points at infinity.

Explanation

In the projective plane, algebraic curves are defined by homogeneous polynomials. Bézout’s Theorem asserts that the total number of intersection points between two such curves depends solely on their degrees — provided they do not share a common factor (i.e. no overlapping components). This count includes:

Points at infinity (thanks to projective coordinates)
Complex solutions (even if not visible in real geometry)
Multiplicities (e.g. tangency counts as multiple intersections)

This principle is foundational in algebraic geometry and underpins many constructions in the Encyclopedia of Poly Geometry (EPG), especially those involving constraints, uniqueness, and curve determination.

Properties Derived from Bézout’s Theorem

Intersection with a Line
A simple consequence of Bézout’s Theorem is that any polynomial curve of degree n intersects a line in exactly n points — counted with multiplicity. This applies even when some of those points lie at infinity or coincide.
Maximum Number of Intersections
Two curves of degrees m and n can intersect in at most m · n points.
For example:
- A line (m = 1) and a conic (n = 2) intersect in at most 2 points.
- Two cubics (m = n = 3) intersect in at most 9 points.
This maximum is achieved when the curves are in general position and share no common component.
Determination via Point Constraints
If a curve of degree n is required to pass through m · n points lying on another curve of degree m, then — under generic conditions — it must share a component with that curve. This is a powerful tool for proving uniqueness, dependency, or forced overlap in geometric constructions.
Tangency and Multiplicity
If two curves touch at a point (i.e. are tangent), that point counts with multiplicity greater than one.
For instance, if a line is tangent to a conic, Bézout still counts two intersections — one with multiplicity two. This allows algebraic tracking of geometric behavior such as contact order and inflection.

Polynomial Curves

Polynomial Curves in Homogeneous Coordinates

Polynomial curves are smooth graphs defined by equations involving powers of variables. In homogeneous coordinates, we use (x : y : z) instead of (x, y), making the curve scale-invariant and allowing us to include points at infinity.

Why Homogeneous Coordinates?

Polynomial curves are the graphs of functions built from powers of variables. Rather than using standard coordinates (x, y), we switch to homogeneous coordinates (x : y : z) to embed the curve in projective space. This makes the description invariant under scaling and includes points “at infinity” that are otherwise excluded in affine geometry.

For example, the parabola y = x² becomes the projective curve yz = x², a homogeneous equation of degree 2.

Polynomial Curves in Projective Space

Polynomial curves in projective geometry are defined using homogeneous coordinates, which offer a scale-invariant way to represent points. Instead of standard Cartesian coordinates (x, y), we use triples (x : y : z), where scalar multiples represent the same point: (x : y : z) ∼ (λx : λy : λz) for any nonzero λ. This formulation naturally includes points at infinity and supports projective transformations.

There are multiple ways to arrive at homogeneous coordinates:

Affine Extension: Starting from Cartesian coordinates, we introduce a third variable z to homogenize polynomial equations. For example, the affine parabola y = x² becomes yz = x², a homogeneous equation of degree 2 in projective space ℙ².
Barycentric Coordinates: Homogeneous coordinates also arise in triangle-based systems, where a point is expressed relative to three reference points A, B, and C of a triangle. A point P is given by a triple (α : β : γ), representing weighted contributions from each vertex. These coordinates are homogeneous because scaling all weights by the same factor does not change the represented point. Barycentric coordinates are especially useful in interpolation, triangle geometry, and invariant descriptions of geometric loci.

Both approaches yield homogeneous representations, but they reflect different geometric intuitions: one rooted in algebraic extension, the other in relational geometry. Together, they underscore the versatility of homogeneous systems in describing curves, configurations, and transformations across affine and projective settings.

A homogeneous polynomial equation of degree n is expressed in variables x, y, and z, such that in every term, the sum of the exponents of x, y, and z equals n. This condition ensures that the equation remains invariant under scaling and properly defines a curve in projective space.

Exploratory Questions

What properties characterize polynomial curves for a specific n ?
What properties characterize polynomial curves for general n ?
Is there a general method for constructing polynomial curves given n(3+n)/2 points ?

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