Analytic geometry creates a connection between graphs and equations. For example, the linear function f(x) = x2 – 2 (an equation) can also be represented by a graph:
Euclidean Geometry is based solely on geometric axioms without formulas or co-ordinates; Analytic geometry is the “marriage” of algebra and geometry with axes and co-ordinates .
Calculus and Analytic Geometry
Calculus and analytic geometry have become so intertwined, it’s rare nowadays to find a course in pure “Analytic Geometry”. It’s more common to take a course in Calculus and Analytic Geometry, which blends the principles of basic analytic geometry with concepts like functions, limits, continuity, derivatives, antiderivatives, and definite integrals.
Topics in Analytic Geometry A to Z
- Arc Length Formula: An “arc” is a curve segment; The arc length formula tells you how long this segment is.
- Area of a Bounded Region: are of a shape contained within a set of functions.
- Area under the curve: Calculating the area between a graph and the x-axis.
- Centroid: The average of all points in an object (e.g. the center of volume or mass).
- Bipolar Coordinate System: An unusual system used in GIS and electrical field theory.
- Center function: gives the trilinear coordinates of a triangle’s center.
- Coterminal Angles: Angles that have the same terminal side.
- Distance Formula / Function: measures the distance between two points in a set (e.g. on a line).
- Delta x / Delta y: Distance traveled along the x- or y-axis.
- Displacement Function: gives us how far a particle has moved from a starting point at an given time.
- Distance Traveled (using derivatives).
- Double Angle Formulas: Sin, Cos, Tan
- Epicycles of Ptolemy: A ancient model of the universe.
- Hyperbola: Two symmetrical curves that have special properties.
- Intersection of lines: The place where two or more graphs cross each other.
- Length of a Line Segment: Measuring “how far” along an x or y axis.
- Level Curve: Analogous to contour lines on a map.
- Parabola: a u-shaped curve; The graph of a quadratic function.
- Parallel Cross Sections: repeated cross sections for a solid, parallel to each other.
- Pedal Coordinates: Tangential coordinates.
- Polar coordinates: “Circular” coordinates on a plane.
- Rate of change: a measure (a rate) of how things are changing.
- Slope: the ratio of a change in x (δx) to a change in y (δy).
- Quadrant: one of the four regions of the Cartesian plane / x-y axis.
- Riemann Sums: Estimating the area under a curve with rectangles.
- Secant line: A secant line connects two ore more points on a curve; An external secant is the “outside” part of the secant line.
- Sketching Graphs on the Cartesian Plane.
- Special Curves: Definition, Examples
- Spherical coordinates: Coordinate system on a sphere.
- Tangent line: a line that touches a graph at only one point and is practically parallel. See also: Vertical Tangents and Horizontal Tangents.
- Tautochrone Problem / Brachistochrone: Classic problems about swinging pendulums.
- Testing for Symmetry of a Function.
- Transformations: shifts, dilations and other “movement” along the x or y axis.
- Trilinear Coordinates: a way to represent a point in the plane using a triangle as the coordinate system.
- Vectors: show magnitude and direction.
- Velocity: Rate of change of displacement.
- x, y coordinate system: A system with a horizontal (x) axis and vertical (y) axis.
- x and y intercepts: The points where a graph crosses the x-axis or y-axis.
- X Y Plane
- Equiangular Spiral, Spira Mirabilis (Logarithmic Spiral)
- Lamé curve
- Quadric Surface: Equations, Examples
In 3D space (also called xyz space), the xy plane contains the x-axis and y-axis:
The xy plane can be described as the set of all points (x, y, z) where z = 0. In other words, any point (x, y, 0). For example, all of the following points are on the xy plane:
- (1, 5, 0)
- (-2, 19, 0)
- (π, -1, 0)
- (.5, .2, 0)
This fact gives us the equation for the xy plane: z = 0.
This is just an extension of the same idea of the x-axis (in the Cartesian plane) being the place where y = 0:
The xy plane, together with the yz plane and xz plane, divide space into eight octants. The O in the center of the diagram is the origin, which is a starting point for the 3D-coordinate system. The points are described by an ordered triple of real numbers (x, y, z). For example, the point (2, 3, 0) can be found at:
- x = 2,
- y = 3,
- z = 0.
As z is zero, we know this point must be somewhere on the xy plane.
Distance Formula for Points in the XY Plane
The distance between any two points in xyz-space can be found with a generalization of the distance formula:
Example question: What is the distance between the points (4, 3, 0) and (2, 9, 0)?
Step 1: Identify the coordinate components that we need to put into the formula. We know our coordinates are always ordered (x, y, z), so:
- (4, 3, 0):
- x1 = 4
- y1 = 3
- z1 = 0.
- (2, 9, 0):
- x2 = 2
- y2 = 9
- zz = 0.
Don’t worry about which coordinate is which (e.g. does x = 4 go into x1 or x2?). The distance formula squares these values, so you’ll get the same answer no matter which way you choose.
Step 2: Plug your values from Step 1 into the distance formula:
If you aren’t good with algebra, head over to Symbolab and just replace the x, y, z values with your inputs.
 Analytic Geometry and Calculus. Retrieved May 3, 2021 from: math.uci.edu/~ndonalds/math184/analytic.pdf