Analytic Geometry

The distance formula computes the straight-line distance between two points in the coordinate plane, and the midpoint formula finds the point exactly halfway between them. Both are direct consequences of the Pythagorean theorem applied to the horizontal and vertical legs of a right triangle formed by the two points and their coordinate differences. Together they underlie nearly every other formula in analytic geometry — circle equations, conic sections, and vector magnitudes all reduce to the distance formula in disguise.

Three-dimensional coordinates extend the familiar (x, y) plane by adding a third mutually perpendicular axis, z, so every point in space is located by an ordered triple (x, y, z). The three coordinate axes divide space into eight octants, analogous to the four quadrants of the plane. Distance and midpoint formulas generalize directly from two dimensions by adding a z-term, since the underlying logic — the Pythagorean theorem applied one axis at a time — is unchanged; this framework underlies 3D graphics, robotics, physics simulations, and the study of surfaces and solids in space.

In three-dimensional space, a line is determined by a point and a direction vector, while a plane is determined by a point and a normal vector — a single vector perpendicular to every line lying in the plane. This is a step up in subtlety from 2D coordinate geometry: two linear equations in x, y, z (like a plane's equation) no longer pin down a point, they pin down a whole plane, and a line in space is usually described parametrically or as the intersection of two planes rather than by a single equation. The plane, in particular, is the fundamental flat object of 3D geometry, playing the role that the line played in the 2D plane.

A vector equation describes a line or a plane not by an algebraic relation between coordinates, but by 'walking' from a known point along one or more direction vectors. A line is traced out by a point plus a single direction vector, scaled by a free parameter t; a plane is traced out by a point plus two independent direction vectors, scaled by two free parameters s and t. This vector viewpoint is what makes lines and planes easy to describe uniformly in any dimension — the same r(t) = r₀ + tv formula works in the plane, in 3-space, or beyond — and it is the natural language for computer graphics, robotics path-planning, and physics trajectories.

The distance from a point to a line is the length of the shortest path connecting them — which is always the perpendicular segment from the point to the line, never a slanted one. Given a line in the general form Ax + By + C = 0 and a point (x₁, y₁) not on it, a single closed-form formula computes this shortest distance directly from the coefficients and coordinates, without needing to first find the foot of the perpendicular. This formula is a two-dimensional cousin of the point-to-plane distance formula in 3D, and it underlies everything from finding the height of a triangle given its vertices to computing clearances in CAD and robotics.

A locus (plural: loci) is the set of all points satisfying a given geometric condition — the word comes from the Latin for 'place.' Analytic geometry turns a locus problem into algebra: state the condition as an equation relating the coordinates (x, y) of an arbitrary point on the locus, then simplify. This single technique — translate a geometric description into a distance or slope condition, then reduce algebraically — is exactly how the standard equations of circles, parabolas, ellipses, and hyperbolas were originally derived, and it remains the general method for finding the equation of any new geometric locus.

Every conic section (other than a circle centered at the origin) can be written in polar coordinates with one focus at the pole, using a single formula involving the eccentricity e and the semi-latus rectum. This polar form unifies the ellipse, parabola, and hyperbola into one equation, distinguished only by the value of e, and is especially natural for orbital mechanics, since Kepler's laws describe planetary orbits as conics with the Sun at one focus — exactly the pole of the polar coordinate system.

When the general second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0 has a nonzero xy-term (B ≠ 0), the conic it describes is tilted relative to the coordinate axes. Rotating the coordinate system by a suitable angle θ eliminates the xy-term, transforming the equation into the familiar axis-aligned standard form, from which the conic's type and orientation become immediately visible. The rotation angle is determined by cot(2θ) = (A−C)/B, and the discriminant B²−4AC is invariant under rotation, so it classifies the conic's type both before and after rotating.

A conic section (or simply a conic) is a curve obtained by intersecting a plane with the surface of a double cone. The three fundamental types are the ellipse, the parabola, and the hyperbola; the circle is a special case of the ellipse (sometimes historically counted as a fourth type). Ancient Greek mathematicians studied these curves extensively, culminating around 200 BCE in Apollonius of Perga's systematic treatise on their properties. Every conic can also be described algebraically as the solution set of a general second-degree (quadratic) equation in two variables.

A parabola is a plane curve that is mirror-symmetric and approximately U-shaped. It arises as a conic section with eccentricity exactly 1, and equivalently as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix). Parabolas are the graphs of quadratic functions and appear throughout physics as the shape of projectile trajectories.

An ellipse is a plane curve surrounding two focal points such that, for every point on the curve, the sum of the distances to the two foci is constant. It generalizes the circle, which is the special case where the two foci coincide. The elongation of an ellipse is measured by its eccentricity, a number strictly between 0 and 1.

A hyperbola is a smooth plane curve with two mirror-image branches that resemble two infinite bows. It is one of the three conic sections, formed when a plane intersects both nappes of a double cone without passing through the apex. Algebraically, a hyperbola is the set of points where the difference of distances to two fixed foci is constant.

Eccentricity is a single non-negative real number, e, that measures how much a conic section deviates from being circular. It arises naturally from the unified focus-directrix definition of a conic: the ratio of a point's distance to a fixed focus versus its distance to a fixed directrix line is constant along the curve, and that constant ratio is exactly e. Because e alone determines the conic's type — e = 0 a circle, 0 < e < 1 an ellipse, e = 1 a parabola, e > 1 a hyperbola — it functions as a shape 'dial': turning it from 0 upward morphs a circle into increasingly elongated ellipses, then at e = 1 the curve opens up into a parabola, and beyond that into a hyperbola.

The standard-form equations for conics (centered at the origin, vertex at the origin) are the special case where the curve happens to sit at the coordinate system's center. Sliding — translating — a conic to a new center (h, k) is done by the same substitution that shifts any graph: replace x with (x − h) and y with (y − k) everywhere in the standard equation. This turns the clean standard forms into the general translated forms seen throughout applications, and running the substitution in reverse — completing the square on a messy second-degree equation — recovers the center, axes, and eccentricity of a conic given in expanded form.

Every conic section — circle, ellipse, parabola, or hyperbola, in any position or orientation — is the solution set of a single second-degree equation in two variables, Ax² + Bxy + Cy² + Dx + Ey + F = 0. The cross term Bxy is what allows the conic to be tilted relative to the coordinate axes; when B = 0 the axes of the conic are already aligned with the x- and y-axes. Classifying which type of conic a given equation represents comes down to a single number computed from the coefficients — the discriminant B² − 4AC — while the linear terms Dx + Ey shift the conic's center or vertex away from the origin.

A vector in the plane is a quantity with both magnitude and direction, represented as an ordered pair ⟨a, b⟩ or drawn as an arrow from one point to another. Unlike a point, which marks a location, a vector represents displacement, velocity, or force — it can be translated anywhere in the plane without changing its identity, since only its length and direction matter. Vectors support addition, scalar multiplication, and the dot product, and every vector can be decomposed into components along the standard basis vectors i = ⟨1,0⟩ and j = ⟨0,1⟩.

The coordinate plane (Cartesian plane) locates every point in a flat 2D space with a unique pair of numbers (x, y), measured as signed distances along two perpendicular number lines called axes. This bridges algebra and geometry: equations become curves, and curves become equations — the foundation of analytic geometry.

A parametric curve is the actual geometric trace — the set of points (x(t), y(t)) in the plane — produced by a parametric equation, viewed as an object of study in its own right rather than as a computational tool. Where 'parametric equations' emphasizes the algebraic machinery (differentiating, eliminating the parameter, computing arc length), 'parametric curves' emphasizes the resulting shapes: circles and ellipses traced at varying speeds, cycloids traced by a rolling wheel, Lissajous figures, and spirals. The same curve can arise from many different parametrizations, and classic named curves — the cycloid, the astroid, the cardioid — are most naturally defined and studied only in parametric form, since they generally have no simple equation of the form y = f(x).

A polar graph is the set of points satisfying r = f(θ) (or a more general relation between r and θ), plotted by sweeping the angle θ and marking the corresponding radius r at each angle. Because a single point in the plane has infinitely many polar representations — (r, θ), (r, θ+2π), and (-r, θ+π) all name the same point — polar curves can display symmetry that is easy to miss algebraically but leaps out visually: cardioids, roses, and limaçons all owe their elegant petal and heart shapes to simple symmetry tests applied to their defining equations. Checking symmetry before plotting cuts the plotting work roughly in half, since only one 'half' of the curve needs to be traced out point by point.