Analytic Geometry

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 [1].

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

Curves:

• Equiangular Spiral, Spira Mirabilis (Logarithmic Spiral)
• Lamé curve
• Quadric Surface: Equations, Examples

X Y Plane

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:

1. (4, 3, 0):
   • x₁ = 4
   • y₁ = 3
   • z₁ = 0.
2. (2, 9, 0):
   • x₂ = 2
   • y₂ = 9
   • z_z = 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.

References

[1] Analytic Geometry and Calculus. Retrieved May 3, 2021 from: math.uci.edu/~ndonalds/math184/analytic.pdf