Analytic Geometry Cheatsheet

Coordinates

2D Cartesian Coordinates

A point in the plane is represented as, whereis the position on the horizontal axis andis the position on the vertical axis.

Distance Between Two Points in 2D

Given two pointsand, the distancebetween them is:

Polar Coordinates (2D)

In the polar coordinate system, a point in the plane is represented as, where:

• is the distance from the origin to the point (also known as radius or modulus).
• is the angle between the positiveaxis and the line connecting the origin to the point, measured counterclockwise (in radians or degrees).

Conversion Formulas Between Polar and Cartesian:

• From polar to Cartesian:

• From Cartesian to polar:

3D Cartesian Coordinates

A point in three-dimensional space is represented as, whereis the position on the axis perpendicular to theplane.

Cylindrical Coordinates

Cylindrical coordinates are an extension of polar coordinates in three dimensions. A point in this system is expressed as, where:

• is the distance from theaxis to the point (the radius in theplane).
• is the angle in theplane relative to the positiveaxis.
• is the height of the point above theplane (the same as in Cartesian coordinates).

Conversion Formulas Between Cylindrical and Cartesian:

• From cylindrical to Cartesian:

• From Cartesian to cylindrical:

Spherical Coordinates

Spherical coordinates are used to describe points in three-dimensional space using three values:, where:

• is the distance from the origin to the point (also known as radius or modulus).
• is the angle in theplane measured from the positiveaxis (similar to the polar angle in cylindrical coordinates).
• is the angle between the positiveaxis and the line connecting the origin to the point (called the inclination angle or colatitude).

Conversion Formulas Between Spherical and Cartesian:

• From spherical to Cartesian:

• From Cartesian to spherical:

Homogeneous Coordinates

In computer graphics, homogeneous coordinates are used to handle projection transformations more efficiently.

2D Homogeneous

A point in homogeneous coordinates is represented as. To convert from homogeneous to Cartesian:.

3D Homogeneous

A point in 3D homogeneous is represented as. To convert it to Cartesian coordinates:

Points and Distances

Distance Between Two Points

The distancebetween two pointsandin the plane is calculated using the formula:

Midpoint

The midpointof a line segment connecting pointsandis found using:

Length of a Segment

The length of a segment in three-dimensional space between pointsand:

2D Lines

Line Equation

The line equation in slope-intercept form is:

Where:

• is the slope.
• is the-intercept.

General Line Equation

The general form of the line equation is:

where:

Slope of the Line

The slopebetween two pointsandis calculated as:

Line Equation Given Its Slope and a Point

If the slopeand a pointare known, the line equation is:

Equation Given by Two Points

The equation of a line in two dimensions given by two pointsandis:

Parallel Lines

Two lines are parallel if they have the same slope,.

Perpendicular Lines

Two lines are perpendicular if the product of their slopes is -1:

Angle Between Two Lines

If the slopes of two lines areand, the anglebetween them is calculated using:

Distance from a Point to a Line (2D)

For a pointand a line:

Distance Between Two Parallel Lines

For parallel linesand: