Analytic Geometry Simulations

A collection of interactive 3D visualizations and simulations to help you master concepts in analytic geometry.

Introduction to Analytic Geometry - Theory & Concepts

Coordinate systems, distance formula, midpoint formula, slope, angle between lines, symmetry, and area of polygons.

Distance & Slope Explorer

Point 1 (x₁, y₁)

x₁-2

y₁-2

Point 2 (x₂, y₂)

x₂3

y₂4

Distance (d):7.81

Midpoint (M):(0.5, 1)

Slope (m):1.20

The Straight Line - Theory & Concepts

Equations of lines, slopes, intercepts, distances, and angles between lines.

Linear Equation Explorer

Slope (m)1

Y-Intercept (b)0

y = x + 0

Slope: 1

Y-Intercept: (0, 0)

X-Intercept: (0.00, 0)

The Circle - Theory & Concepts

Equations of circles, finding center and radius, tangent lines, and degenerate cases.

Circle Explorer

h (x-center)0

k (y-center)0

r (radius)5

(x - 0)^2 + (y - 0)^2 = 5^2

The Parabola - Theory & Concepts

Equations of parabolas, vertex, focus, directrix, and applications.

Circle Explorer

h (x-center)0

k (y-center)0

r (radius)5

(x - 0)^2 + (y - 0)^2 = 5^2

The Ellipse - Theory & Concepts

Equations of ellipses, finding foci, vertices, eccentricity, and area.

Circle Explorer

h (x-center)0

k (y-center)0

r (radius)5

(x - 0)^2 + (y - 0)^2 = 5^2

The Hyperbola - Theory & Concepts

Equations of hyperbolas, finding foci, vertices, transverse axis, asymptotes, and eccentricity.

Circle Explorer

h (x-center)0

k (y-center)0

r (radius)5

(x - 0)^2 + (y - 0)^2 = 5^2

Polar Coordinates - Theory & Concepts

Polar coordinate system, conversion between rectangular and polar, and graphing polar curves.

Polar Curves

A rose curve. If k=4/1 is integer & odd, it has k petals. If even, 2k petals. If rational, overlapping petals.

r = 2 \cos(4\theta)

Curve Family

Parameter 'a' (2)

Petal factor numerator 'n' (4)

Petal factor denominator 'd' (1)

Polar Grid

Rect. Grid

Translation and Rotation of Axes - Theory & Concepts

Transformation of coordinates through translation and rotation of the axes, and eliminating the xy-term in conic sections.

Transformation of Axes

Translation X (h): 0

Translation Y (k): 0

Rotation Angle (θ): 0°

Parametric Equations - Theory & Concepts

Defining curves using an independent parameter, converting to rectangular form, and applications to conic sections.

Parametric Curve: Ellipse

x = 5 cos(t), y = 3 sin(t)

Curve Type

Param a: 5

Param b: 3

Parameter t: 6.28

Solid Geometry - Theory & Concepts

Three-dimensional coordinate systems, distance, direction cosines, equations of lines and planes in space.

3D Rectangular Coordinates

X Coordinate: 3.0

Y Coordinate: 4.0

Z Coordinate: 5.0

Distance from Origin:

d = 7.07

Quadric Surfaces - Theory & Concepts

Understanding 3D surfaces including spheres, ellipsoids, paraboloids, and hyperboloids.

Quadric Surfaces Explorer

Drag to rotate • Scroll to zoom

Parameters

Param 'a' (X-axis)2.0

Param 'b' (Y-axis)2.0

Param 'c' (Z-axis)2.0

Note: In this 3D view, the vertical axis is actually the Z-axis in standard mathematical notation (mapped to Three.js Y-axis).

Tangents and Normals to Conic Sections - Theory & Concepts

Equations of tangent and normal lines to circles, parabolas, ellipses, and hyperbolas.

Tangent and Normal to a Circle

Point Position Angle: 45°

Observation: The normal line always passes directly through the origin (center of the circle), and the tangent line is always strictly perpendicular to the normal line at the point of contact.