**Analytic geometry** creates a connection between graphs and equations. For example, the linear function f(x) = x^{2} – 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:

- (4, 3, 0):
- x
_{1}= 4 - y
_{1}= 3 - z
_{1}= 0.

- x
- (2, 9, 0):
- x
_{2}= 2 - y
_{2}= 9 - z
_{z}= 0.

- x

Don’t worry about which coordinate is which (e.g. does x = 4 go into x_{1} or x_{2}?). 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.

## Quadric Surface

A **quadric surface** is a 3D extension of a conic (ellipsis, hyperbola, or parabola). In 2D space, they are defined by quadratic equations in 2D space [2]. *Quadric *refers to the degree of the equation describing the surface: variables in these equations are raised to the 2nd power. They can be described, in general, as a graph of an equation expressed in the form [3]:

Ax^{2} + B^{2} + C^{2} + Dxy + Exz + Fyz + Hx + Iy + Jz + K = 0.

Where A, B, C, D, E, F, H, I, J, K are fixed constants and x, y, z, are variables.

Different types of quadric surfaces can be obtained from simplifying this equation by rotations and translations of the x, y, z coordinate axes.

## Types of Quadric Surfaces

There are six basic types of quadric surface [2]:

- Ellipsoid,
- Elliptic paraboloids,
- Hyperbolic paraboloid,
- Cones,
- Hyperboloids of one sheet,
- Hyperboloids of two sheets.

Other examples of quadratic surfaces include the cylinder, elliptic cone, elliptic cylinder, elliptic hyperboloid, hyperbolic cylinder, paraboloid, sphere, and spheroid.

**1. Ellipsoid**

The general equation for an ellipsoid is:

If A = B = C, the shape is a sphere.

**2. Elliptic paraboloids**

The equation for an elliptic paraboloid is:

Elliptic paraboloids have ellipses as cross sections. If A = B, the cross section is a circle.

**3. Hyperbolic paraboloid**

The equation for a hyperbolic paraboloid is:

Hyperbolic paraboloids are saddle-shaped, like a Pringle. The sign of c determines whether the graph opens up or down. A positive value for c has the saddle right side up (as if it were being placed on a horse).

**4. Cones**

The equation for a cone is [4]:

**5. Hyperboloids of one sheet**

The equation for the hyperboloid of one sheet is:

**6. Hyperboloids of two sheets**

The equation for the hyperboloid of two sheets is:

## References

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

[2] Quadric Surfaces. Retrieved July 31, 2021 from: https://web.cs.wpi.edu/~matt/courses/cs563/talks/renderman/quadric.html

[3] Quadric Surfaces. Retrieved August 1, 2021 from: https://opentextbc.ca/calculusv3openstax/chapter/quadric-surfaces/

[4] Quadratic Surfaces. Retrieved August 8, 2021 from: http://www.staff.city.ac.uk/o.castro-alvaredo/teaching/surfaces.pdf