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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 - Analytic Geometry Distance Slope

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 Formula:
d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
d=(3−(−2))2+(4−(−2))2=(5)2+(6)2≈7.81d = \sqrt{(3 - (-2))^2 + (4 - (-2))^2} = \sqrt{(5)^2 + (6)^2} \approx 7.81
Slope Formula:
m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}
m=4−(−2)3−(−2)=65≈1.20m = \frac{4 - (-2)}{3 - (-2)} = \frac{6}{5} \approx 1.20
Midpoint Formula:
M=(x1+x22,y1+y22)M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
M=(−2+32,−2+42)=(0.5,1)M = \left(\frac{-2 + 3}{2}, \frac{-2 + 4}{2}\right) = (0.5, 1)
Coordinate system graph displaying distance and slope between two pointsGrid line patternGrid area backgroundX AxisY AxisConnection segment of length 7.81Slope triangle: Run = 5, Rise = 6Midpoint M at (0.5, 1)MPoint 1: (-2, -2)P₁Point 2: (3, 4)P₂

Introduction to Analytic Geometry - Theory & Concepts - Analytic Geometry Section Formula

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

Section Formula Explorer (Line Segment Division)

Parameters

Ratio Factor m2
Ratio Factor n1

Step-by-Step Solver

Governing Formulas
x=mx2+nx1m+nx = \frac{m x_2 + n x_1}{m + n}y=my2+ny1m+ny = \frac{m y_2 + n y_1}{m + n}
Substitution & Result
x=(2)(6)+(1)(−6)2+1=12+−63=2.00x = \frac{(2)(6) + (1)(-6)}{2 + 1} = \frac{12 + -6}{3} = 2.00
y=(2)(4)+(1)(−4)2+1=8+−43=1.33y = \frac{(2)(4) + (1)(-4)}{2 + 1} = \frac{8 + -4}{3} = 1.33
Section Point: Point P divides segment AB in the ratio 2 : 1. The resulting coordinate is P(2.00, 1.33).
-10 to 10Point A (-6, -4)A(-6, -4)Point B (6, 4)B(6, 4)Point P (2.00, 1.33)P(2.0, 1.3)

Drag points A and B on the coordinate system to recalculate

Introduction to Analytic Geometry - Theory & Concepts - Analytic Geometry Polygon Area

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

Polygon Area Explorer (Shoelace Formula)

Vertices

Vertex P1(1, 2)
X-Coord
Y-Coord
Vertex P2(3, 5)
X-Coord
Y-Coord
Vertex P3(4, 0)
X-Coord
Y-Coord
Calculated Area
6.5 sq units
Grid displaying polygon with verticesGrid line patternGrid area backgroundX AxisY AxisPolygon with area 6.5Vertex P1 at (1, 2)P1Vertex P2 at (3, 5)P2Vertex P3 at (4, 0)P3
Shoelace Formula Calculation:
A=12∣∑i=1n(xiyi+1−xi+1yi)∣A = \frac{1}{2} \left| \sum_{i=1}^n (x_i y_{i+1} - x_{i+1} y_i) \right|
A=12∣[(1⋅5)+(3⋅0)+(4⋅2)]−[(2⋅3)+(5⋅4)+(0⋅1)]∣A = \frac{1}{2} \left| [(1 \cdot 5) + (3 \cdot 0) + (4 \cdot 2)] - [(2 \cdot 3) + (5 \cdot 4) + (0 \cdot 1)] \right|A=12∣13−26∣=12∣−13∣=6.5A = \frac{1}{2} \left| 13 - 26 \right| = \frac{1}{2} \left| -13 \right| = 6.5

Introduction to Analytic Geometry - Theory & Concepts - Analytic Geometry Conic Section Cutter

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

Conic Section Cutter

Result: Circle

The cutting plane is parallel to the base of the cone.

x2+y2=r2x^2 + y^2 = r^2
0
0
Cone Side Angle: 56.3°
Cone AxisGeneratrixCutting Plane

The Straight Line - Theory & Concepts - Analytic Geometry Line

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

Linear Equation Explorer

1.0
0.0
y=x+0y = x + 0

Interactive Insights

Line Slope (m):Rising (Positive)
Y-Intercept (0, b):(0, 0.0)
X-Intercept (-b/m, 0):(0.00, 0)
Graph of line y = x + 0Grid lineGrid backgroundHorizontal AxisVertical AxisLine equation: y = x + 0Y Intercept at (0, 0.0)X Intercept at (0.00, 0)

The Straight Line - Theory & Concepts - Analytic Geometry Parallel Perpendicular

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

Parallel & Perpendicular Line Explorer

Base Line Controls

Coefficient A2
Coefficient B3
Constant C-4

Draggable Point P

P_x Coordinate
P_y Coordinate

Calculated Equations

Base Line (L₁)
2x+3y−4=02x + 3y - 4 = 0
Slope (m₁):-0.67
Parallel Line through P (Lₚ)
2x+3y−18=02x + 3y - 18 = 0
Slope (mₚ):-0.67
Perpendicular Line through P (L⟂)
3x−2y−1=03x -2y - 1 = 0
Slope (m⟂):1.50
Point P (3, 4)P(3, 4)

Drag point P on the coordinate system to shift parallel and perpendicular lines

The Straight Line - Theory & Concepts - Analytic Geometry Line Distance Angle

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

3.0
4.0
-12.0
4.0
5.0
Calculated Distanced = 4.000
Shortest distance from Point P (4, 5) to line is 4.000Grid lineGrid backgroundHorizontal AxisVertical AxisLine Equation: 3x + 4y + -12 = 0Perpendicular projection vector from P to the linePoint P at (4.0, 5.0)Closest Point Q on line at (1.6, 1.8)
d=∣(3.0)(4.0)+(4.0)(5.0)+(−12.0)∣(3.0)2+(4.0)2=4.000d = \frac{|(3.0)(4.0) + (4.0)(5.0) + (-12.0)|}{\sqrt{(3.0)^2 + (4.0)^2}} = 4.000

The Circle - Theory & Concepts - Analytic Geometry Conic Sections

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

Circle Explorer

0
0
5
(x−0)2+(y−0)2=52(x - 0)^2 + (y - 0)^2 = 5^2

Interactive Insights

Center Coordinate (h, k):(0, 0)
Radius (r):5 units
Area:78.54 units²
Circumference:31.42 units
Graph showing dynamic conic section of type circleGrid lineGrid backgroundHorizontal AxisVertical AxisCircle with radius 5 centered at (0, 0)Center at (0, 0)

The Circle - Theory & Concepts - Analytic Geometry Three Point Circle

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

Circle Passing Through Three Points

Point Coordinates

Point A
(-4.0, 3.0)
Point B
(5.0, 4.0)
Point C
(1.0, -4.0)
Matrix System
(−4315411−41)(DEF)=(−25−41−17)\begin{pmatrix} -4 & 3 & 1 \\ 5 & 4 & 1 \\ 1 & -4 & 1 \end{pmatrix} \begin{pmatrix} D \\ E \\ F \end{pmatrix} = \begin{pmatrix} -25 \\ -41 \\ -17 \end{pmatrix}
Cramer's Rule Determinants
det(M) = -68.00
det(D) = 104.00
det(E) = 152.00
det(F) = 1660.00
D = det(D)/det = -1.53, E = det(E)/det = -2.24, F = det(F)/det = -24.41
General Equation
x2+y2−1.53x−2.24y−24.41=0x^2 + y^2 - 1.53x - 2.24y - 24.41 = 0
Standard Equation
(x−0.76)2+(y−1.12)2=26.25(x - 0.76)^2 + (y - 1.12)^2 = 26.25
C(0.8, 1.1)Point A (-4, 3)A(-4, 3)Point B (5, 4)B(5, 4)Point C (1, -4)C(1, -4)

Drag points A, B, and C to dynamically recalculate the circumscribed circle

The Parabola - Theory & Concepts - Analytic Geometry Parabola Focus Directrix

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

Parabola: Focus & Directrix Explorer

Parabola parameters

Vertex h (x-center)0
Vertex k (y-center)-2
Focal parameter p (focus distance)2

Test point on parabola

Point P x-coordinate (x_P)4.0

Geometric Locus Distance Verification

Equation & Coordinates
(x−h)2=4p(y−k)(x - h)^2 = 4p(y - k)
Focus: F(h,k+p)=F(0,0.0)F(h, k + p) = F(0, 0.0)
Directrix: y=k−p  ⟹  y=−4.0y = k - p \implies y = -4.0
Point: P(xP,yP)=P(4.0,0.00)P(x_P, y_P) = P(4.0, 0.00)
1. Distance to Focus (d₁)
d1=(xP−h)2+(yP−yF)2=(4.0−0)2+(0.00−0.0)2=4.000d_1 = \sqrt{(x_P - h)^2 + (y_P - y_F)^2} = \sqrt{(4.0 - 0)^2 + (0.00 - 0.0)^2} = 4.000
2. Distance to Directrix (d₂)
d2=∣yP−ydir∣=∣0.00−(−4.0)∣=4.000d_2 = |y_P - y_{\text{dir}}| = |0.00 - (-4.0)| = 4.000
d₁ = d₂ = 4.000 units (Verified!)
DIRECTRIX (y = -4.0)Focus F (0, 0)F(0, 0.0)Vertex V (0, -2)V(0, -2)Point P (4.0, 0.0)P(4.0, 0.0)

Move the "Point P x-coordinate" slider on the left to trace the parabola

The Ellipse - Theory & Concepts - Analytic Geometry Ellipse Foci Construction

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

Ellipse: Foci String Construction Explorer

Geometric parameters

Semi-major axis a (string half-length)6
Focal distance c (focus location)4

Tracing Angle

Angle θ (theta)45°

Construction verification

Ellipse Formula
x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
a=6.0, b=a2−c2=62−42=4.47a = 6.0, \ b = \sqrt{a^2 - c^2} = \sqrt{6^2 - 4^2} = 4.47
Foci: F1(−4,0), F2(4,0)F_1(-4, 0), \ F_2(4, 0)
Distances to Foci
d1=PF1=8.828, d2=PF2=3.172d_1 = PF_1 = 8.828, \ d_2 = PF_2 = 3.172
Pins-and-String Sum
d1+d2=8.828+3.172=12.0=2ad_1 + d_2 = 8.828 + 3.172 = 12.0 = 2a
The total string length remains exactly 12 units at all angles.
Focus F1 (-4, 0)F₁Focus F2 (4, 0)F₂P (4.2, 3.2)P(4.2, 3.2)

Move the "Angle θ" slider to trace the ellipse. The indigo string adjusts dynamically.

The Hyperbola - Theory & Concepts - Analytic Geometry Hyperbola Asymptote Explorer

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

Hyperbola & Asymptote Explorer

Hyperbola parameters

Semi-transverse axis a3.5
Semi-conjugate axis b2.5

Test point on hyperbola

Point P y-coordinate (y_P)2.0

Verification details

Hyperbola Standard Equation
x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1
a=3.5, b=2.5, c=a2+b2=4.30a = 3.5, \ b = 2.5, \ c = \sqrt{a^2 + b^2} = 4.30
Asymptotes: y=±bax=±0.71xy = \pm \frac{b}{a}x = \pm 0.71x
Distances to Foci
d1=PF1=9.008, d2=PF2=2.008d_1 = PF_1 = 9.008, \ d_2 = PF_2 = 2.008
Hyperbola Locus Metric
∣d1−d2∣=∣9.008−2.008∣=7.0=2a|d_1 - d_2| = |9.008 - 2.008| = 7.0 = 2a
The focal distance difference is constant and equals the transverse axis length 7.
Focus F1 (-4.30, 0)F₁Focus F2 (4.30, 0)F₂Point P (4.5, 2.0)P(4.5, 2.0)

The asymptotes act as boundary lines (dashed gray) which the hyperbola approaches as $x$ increases

Polar Coordinates - Theory & Concepts - Analytic Geometry Polar Coordinates

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

Polar Coordinates & Curves Explorer

Graph equations defined by $r = f(\theta)$ in the polar coordinate system

r=2cos⁡(4θ)r = 2 \cos(4\theta)
Description

A rose curve. If k=4.00 is an integer and odd, it has 4 petals. If even, 8 petals. If rational, overlapping petals.

2
4
1

Polar Coordinates - Theory & Concepts - Analytic Geometry Polar Cartesian Conversion

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

Polar-Cartesian Conversion Visualizer

Interactively convert between rectangular coordinates $(x, y)$ and polar coordinates $(r, \theta)$

X-Coordinate4.00
Y-Coordinate3.00
Show Polar Grid Overlay
Coordinate Summary
Cartesian (x, y)(4.00, 3.00)
Polar (r, θ)(5.00, 36.9°)
Point P
Radius Conversion Formula ($r$)r=x2+y2=(4.00)2+(3.00)2=5.000r = \sqrt{x^2 + y^2} = \sqrt{(4.00)^2 + (3.00)^2} = 5.000
Angle Conversion Formula ($\\theta$)θ=atan2(y,x)=tan⁡−1(3.004.00)=36.87∘\theta = \text{atan2}(y, x) = \tan^{-1}\left(\frac{3.00}{4.00}\right) = 36.87^\circ

Polar Coordinates - Theory & Concepts - Analytic Geometry Polar Curves Rose Limacon

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

Polar Curves: Rose & Limaçon Explorer

Parameters

Parameter a (Base scale)5
Petal frequency integer n5

If $n$ is odd, the rose has $n$ petals. If $n$ is even, it has $2n$ petals.

Tracing Control

Angle Limit (θ_max)360°

Dynamic coordinate solver

Polar Equation
r=acos⁡(nθ)  ⟹  r=5.0cos⁡(5θ)r = a \cos(n \theta) \implies r = 5.0 \cos(5 \theta)
Step 1: Current Radius
θ=360∘ (6.283 rad)  ⟹  r=5.000\theta = 360^{\circ} \ ( 6.283 \text{ rad} ) \implies r = 5.000
Step 2: Polar to Cartesian conversion
x=rcos⁡(θ)=5.00cos⁡(360∘)=5.00y=rsin⁡(θ)=5.00sin⁡(360∘)=−0.00\begin{aligned} x &= r \cos(\theta) = 5.00 \cos(360^{\circ}) = 5.00 \\ y &= r \sin(\theta) = 5.00 \sin(360^{\circ}) = -0.00 \end{aligned}
Point P (r=5.00, θ=360°)

The dashed line shows the completed shape. The blue line traces the curve up to θ = 360°

Translation and Rotation of Axes - Theory & Concepts - Analytic Geometry Transformation

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

Transformation of Axes

Translate and rotate coordinate axes, mapping point P(x,y) to P'(x',y')

0.0
0.0
0°
Governing Equation
{x′=(x−h)cos⁡θ+(y−k)sin⁡θy′=−(x−h)sin⁡θ+(y−k)cos⁡θ\begin{cases} x' = (x - h)\cos\theta + (y - k)\sin\theta \\ y' = -(x - h)\sin\theta + (y - k)\cos\theta \end{cases}
Transformed Coordinates
P′(x′,y′)=P′(3.00,4.00)P'(x', y') = P'(3.00, 4.00)
Visual Guide:The original coordinate axes are drawn in light gray/slate, while the transformed (translated by h, k and rotated by θ) coordinate axes are shown in red. Point P(3, 4) is mapped into its new coordinate system values P'.

Translation and Rotation of Axes - Theory & Concepts - Analytic Geometry Axes Rotation Matrix Explorer

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

Rotation of Axes Matrix Explorer

Rotation Control

Rotation Angle θ30°

Draggable point coordinates

Original Coordinates (x, y)
(5.0, 4.0)
Rotated Coordinates (x', y')
(6.33, 0.96)

Transformation Matrix solver

Rotation Transformation Matrix
(x′y′)=(cos⁡θsin⁡θ−sin⁡θcos⁡θ)(xy)\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}
Step 1: Trigonometric Values
θ=30∘  ⟹  cos⁡θ=0.866, sin⁡θ=0.500\theta = 30^{\circ} \implies \cos\theta = 0.866, \ \sin\theta = 0.500
Step 2: Linear Substitution
x′=5⋅(0.866)+4⋅(0.500)=6.33x' = 5 \cdot (0.866) + 4 \cdot (0.500) = 6.33
y′=−5⋅(0.500)+4⋅(0.866)=0.96y' = -5 \cdot (0.500) + 4 \cdot (0.866) = 0.96
xyx'y'Point P (5.0, 4.0)P(5.0, 4.0)

Drag point P or adjust angle θ to see original vs. rotated projection lines

Translation and Rotation of Axes - Theory & Concepts - Analytic Geometry Curve Transformation

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

2.0
-1.0

Standard Ellipse (h=0, k=0):

x²/9 + y²/4 = 1

Translating axes shifts standard coordinates to: (x - h) and (y - k).

Shifted ellipse centered at (2, -1)Grid lineGrid backgroundOriginal Horizontal AxisOriginal Vertical AxisTranslated Horizontal AxisTranslated Vertical AxisShifted Ellipse CurveCenter Point at (2, -1)
(x−2.0)29+(y−(−1.0))24=1\frac{(x - 2.0)^2}{9} + \frac{(y - (-1.0))^2}{4} = 1

Parametric Equations - Theory & Concepts - Analytic Geometry Parametric

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

Ellipse Parametric Curve

Visualize equations in parametric form

x(t)=5cos⁡(t),y(t)=3sin⁡(t)x(t) = 5 \cos(t), \quad y(t) = 3 \sin(t)
Parameter a5
Parameter b3
Parameter t
6.28 rad / 360°

Parametric Equations - Theory & Concepts - Analytic Geometry Projectile Parametric

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

Parametric Trajectory Explorer

Visualize projectile motion using 2D parametric equations

45°
12 m/s
9.8 m/s²
Time t = 0.00s
x(t) = 0.00 m
y(t) = 0.00 m
{x(t)=(8.5)ty(t)=(8.5)t−12(9.8)t2\begin{cases} x(t) = (8.5) t \\ y(t) = (8.5) t - \frac{1}{2} (9.8) t^2 \end{cases}

Parametric Equations - Theory & Concepts - Analytic Geometry Cycloid Epicycloid Visualizer

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

Cycloid & Epicycloid Tracer

Rolling Radius r0.80
Parameter θ (Angle)0°
Parametric Equations
x(θ)=r(θ−sin⁡θ)=0.80(θ−sin⁡θ)y(θ)=r(1−cos⁡θ)=0.80(1−cos⁡θ)\begin{aligned} x(\theta) &= r(\theta - \sin\theta) = 0.80(\theta - \sin\theta) \\ y(\theta) &= r(1 - \cos\theta) = 0.80(1 - \cos\theta) \end{aligned}

Solid Geometry - Theory & Concepts - Analytic Geometry Solid Geometry

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

3D Rectangular Coordinates

Visualize coordinates $(x, y, z)$ and distance to the origin in three-dimensional space

3.0
4.0
5.0
d=x2+y2+z2=(3.0)2+(4.0)2+(5.0)2=7.07d = \sqrt{x^2 + y^2 + z^2} = \sqrt{(3.0)^2 + (4.0)^2 + (5.0)^2} = 7.07
Interactive Insights
Coordinates (x, y, z):(3.0, 4.0, 5.0)
Dist. to Origin (d):7.07 units
Octant Location:Octant I (All +)
Drag to rotate • Scroll to zoom

Solid Geometry - Theory & Concepts - Analytic Geometry Plane Three Points3 D

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

Plane Defined by Three Points

x₁2.0
y₁1.0
z₁3.0
x₂-1.0
y₂3.0
z₂1.0
x₃1.0
y₃-2.0
z₃-1.0
Drag to rotate view • Scroll to zoom
1. Direction Vectors
v⃗=P1P2→=⟨−3.0,2.0,−2.0⟩\vec{v} = \overrightarrow{P_1P_2} = \langle -3.0, 2.0, -2.0 \ranglew⃗=P1P3→=⟨−1.0,−3.0,−4.0⟩\vec{w} = \overrightarrow{P_1P_3} = \langle -1.0, -3.0, -4.0 \rangle
2. Normal Vector
n⃗=v⃗×w⃗=⟨A,B,C⟩\vec{n} = \vec{v} \times \vec{w} = \langle A, B, C \ranglen⃗=⟨−14.0,−10.0,11.0⟩\vec{n} = \langle -14.0, -10.0, 11.0 \rangle
3. Plane Equation
A(x−x1)+B(y−y1)+C(z−z1)=0A(x - x_1) + B(y - y_1) + C(z - z_1) = 0
gives:
(−14.0)x+(−10.0)y+(11.0)z+(5.0)=0(-14.0)x + (-10.0)y + (11.0)z + (5.0) = 0

Solid Geometry - Theory & Concepts - Analytic Geometry Line Plane3 D

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

Plane: Ax + By + Cz + D = 0

Line Parameter t

Intersection Point
I = (7.00, 5.00, -5.00)
(1.0)x+(1.0)y+(2.0)z+(−2.0)=0(1.0)x + (1.0)y + (2.0)z + (-2.0) = 0Line: r⃗(t)=⟨−2.0,−1.0,4.0⟩+t⟨1.5,1.0,−1.5⟩\text{Line: } \vec{r}(t) = \langle -2.0, -1.0, 4.0 \rangle + t \langle 1.5, 1.0, -1.5 \rangle

Solid Geometry - Theory & Concepts - Analytic Geometry Cylindrical Spherical3 D

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

Conversions
Cartesian (x,y,z):(3.0, 3.0, 3.0)
Cylindrical (r,θ,z):(4.2, 45°, 3.0)
Spherical (ρ,θ,φ):(5.2, 45°, 55°)
r=x2+y2=4.24,ρ=x2+y2+z2=5.20r = \sqrt{x^2+y^2} = 4.24, \quad \rho = \sqrt{x^2+y^2+z^2} = 5.20

Quadric Surfaces - Theory & Concepts - Analytic Geometry Quadric Surfaces

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

Quadric Surfaces Explorer

Currently displaying a 3D surface of type ellipsoid with X-axis parameter a set to 2.0, Y-axis parameter b set to 2.0, and Z-axis parameter c set to 2.0.
Drag to rotate • Scroll to zoom
2.0
2.0
2.0
x222+y222+z222=1\frac{x^2}{2^2} + \frac{y^2}{2^2} + \frac{z^2}{2^2} = 1
Note: In this 3D view, the vertical axis is actually the Z-axis in standard mathematical notation (mapped to Three.js Y-axis).

Quadric Surfaces - Theory & Concepts - Analytic Geometry Quadric Surface Slicer

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

Quadric Surface Slicer

Slice 3D quadric surfaces with a horizontal plane and visualize the 2D intersection curve

1.00
Surface Dimensions
3.0
2.5
2.0
Intersection Result
Ellipse with Rₓ = 2.60, Rᵧ = 2.17
x29.0+y26.3+z24.0=1∩z=1.00\frac{x^2}{9.0} + \frac{y^2}{6.3} + \frac{z^2}{4.0} = 1 \quad \cap \quad z = 1.00

Quadric Surfaces - Theory & Concepts - Analytic Geometry Quadric Slices3 D

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

Quadric Surface Intersection Slices

Constant c1.00
Surface Dimensions
Parameter a1.5
Parameter b1.2
3D View (Drag to rotate)
-3-3-2-2-1-1112233XY
2D Slice Projection
Mathematical Equations
z=x22.25+y21.44∩z=1.00z = \frac{x^2}{2.25} + \frac{y^2}{1.44} \quad \cap \quad z = 1.00

Tangents and Normals to Conic Sections - Theory & Concepts - Analytic Geometry Tangent Normal

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

Tangent and Normal to a Circle

Visualize the orthogonal relationship between tangent and normal vectors on a circle

45°
Mathematical Equations
Circle Equation:
x2+y2=1002x^2 + y^2 = 100^2
Tangent Equation:
x⋅(71)+y⋅(71)=1002x \cdot (71) + y \cdot (71) = 100^2
Normal Equation:
(71)y−(71)x=0(71) y - (71) x = 0
Observation: The normal line always passes directly through the origin (center of the circle), and the tangent line is always strictly perpendicular ($90^\circ$) to the normal line at the point of contact.

Tangents and Normals to Conic Sections - Theory & Concepts - Analytic Geometry Conic Tangent Normal

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

Point Position Parameter45
Tangent / Normal Slopes
Point P(x₁, y₁):(2.47, 1.41)
Tangent Slope:-0.571
Normal Slope:1.750
Tangent and Normal line curves for ellipseGrid lineGrid backgroundHorizontal AxisVertical AxisConic curve branchNormal line (perpendicular to tangent)Tangent line (touching curve at point P)Point P of tangency at (2.47, 1.41)
x212.3+y24.0=1\frac{x^2}{12.3} + \frac{y^2}{4.0} = 1

Tangents and Normals to Conic Sections - Theory & Concepts - Analytic Geometry Interactive Tangent Normal Explorer

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

Interactive Tangent & Normal Explorer

Curve Factor c0.50
Point of Tangency x₀1.50
Point P:(1.50, -0.88)
Tangent Slope (m_t):1.500
Normal Slope (m_n):-0.667
Slope Product:-1.00
Click or drag on canvas to position P
Tangent Liney=1.50x+(−3.13)y = 1.50x + (-3.13)
Normal Liney=−0.67x+(0.13)y = -0.67x + (0.13)
Slope Perpendicularity Identity: mtangent⋅mnormal=−1m_{tangent} \cdot m_{normal} = -1

Vector Analytic Geometry - Theory & Concepts - Vector Resolution3 D

Introduction to vectors, dot product, cross product, and their applications in analytic geometry.

3D Vector Configuration

Adjust the magnitude and two coordinate direction angles. The third angle (gamma\\gamma) is calculated automatically to satisfy the identity cos2alpha+cos2beta+cos2gamma=1\\cos^2\\alpha + \\cos^2\\beta + \\cos^2\\gamma = 1.

100 N
60°
60°
Calculated Angle γ (from z-axis):
γ≈45.0∘\gamma \approx 45.0^\circ

Rectangular Components

X - Component (FxF_x)+50.0 N
Y - Component (FyF_y)+50.0 N
Z - Component (FzF_z)+70.7 N
Cartesian Vector Formulation
F={50.0i+50.0j+70.7k} N\mathbf{F} = \{50.0\mathbf{i} + 50.0\mathbf{j} + 70.7\mathbf{k}\}\text{ N}

Vector Analytic Geometry - Theory & Concepts - Analytic Geometry Vector Projection Angle

Introduction to vectors, dot product, cross product, and their applications in analytic geometry.

Vector Projection & Angle Explorer

Vector u = ⟨u_x, u_y⟩
u_x3.0
u_y2.0
Vector v = ⟨v_x, v_y⟩
v_x4.0
v_y0.0
Dot Product (u · v):12.00
Angle (θ):33.7°
Projection Scalar:0.750
proj_v u:⟨3.00, 0.00⟩
Vector uVector vProjection Vectoruvproj_v u
Click: change u • Shift+Click: change v
Mathematical Representation
u⃗⋅v⃗=uxvx+uyvy=(3.0)(4.0)+(2.0)(0.0)=12.00\vec{u} \cdot \vec{v} = u_x v_x + u_y v_y = (3.0)(4.0) + (2.0)(0.0) = 12.00θ=arccos⁡(u⃗⋅v⃗∥u⃗∥∥v⃗∥)=arccos⁡(12.03.6⋅4.0)=33.7∘\theta = \arccos\left(\frac{\vec{u} \cdot \vec{v}}{\|\vec{u}\| \|\vec{v}\|}\right) = \arccos\left(\frac{12.0}{3.6 \cdot 4.0}\right) = 33.7^\circprojv⃗u⃗=(u⃗⋅v⃗∥v⃗∥2)v⃗=(0.75)v⃗=⟨3.00,0.00⟩\text{proj}_{\vec{v}} \vec{u} = \left(\frac{\vec{u} \cdot \vec{v}}{\|\vec{v}\|^2}\right)\vec{v} = (0.75) \vec{v} = \langle 3.00, 0.00 \rangle

Vector Analytic Geometry - Theory & Concepts - Analytic Geometry Vector Product

Introduction to vectors, dot product, cross product, and their applications in analytic geometry.

Vector Dot and Cross Product Visualizer

Manipulate two vectors in 3D space to analyze their dot product, cross product, and spanned parallelogram area

Vector u = ⟨u_x, u_y, u_z⟩
3.0
0.0
3.0
Vector v = ⟨v_x, v_y, v_z⟩
0.0
3.0
2.0
Vector Metrics
Dot Product
6.00
Cross Area
14.07
Angle
66.9°
Drag to rotate • Scroll to zoom
u⃗⋅v⃗=(3.0)(0.0)+(0.0)(3.0)+(3.0)(2.0)=6.00\vec{u} \cdot \vec{v} = (3.0)(0.0) + (0.0)(3.0) + (3.0)(2.0) = 6.00
u⃗×v⃗=⟨−9.0,−6.0,9.0⟩,∣u⃗×v⃗∣=14.07\vec{u} \times \vec{v} = \langle -9.0, -6.0, 9.0 \rangle, \quad |\vec{u} \times \vec{v}| = 14.07

Cylindrical and Spherical Coordinates - Theory & Concepts - Analytic Geometry Coordinate Surfaces3 D

Advanced 3D coordinate systems and their relationship with rectangular coordinates.

Coordinate Surfaces: r=2.0,θ=45∘,z=1.0\text{Coordinate Surfaces: } \quad r = 2.0, \quad \theta = 45^\circ, \quad z = 1.0

Cylindrical and Spherical Coordinates - Theory & Concepts - Analytic Geometry Spherical Volume Element

Advanced 3D coordinate systems and their relationship with rectangular coordinates.

Spherical Volume Element (dV) Explorer

Position Variables
Radius (ρ)3.00
Zenith Angle (φ)45°
Azimuth Angle (θ)60°
Differentials (Size)
Thickness (dρ)0.80
Delta Zenith (dφ)20°
Delta Azimuth (dθ)25°
Approx dV:0.7754
Exact Volume V:1.1533
Difference:32.76%
Drag to rotate view • Scroll to zoom
Spherical Volume Integration
dV=ρ2sin⁡ϕ dρ dϕ dθdV = \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta
Substitution & Calculations
Approx dV≈(3.00)2sin⁡(45∘)⋅(0.80)⋅(0.3491 rad)⋅(0.4363 rad)=0.77543\text{Approx dV} \approx (3.00)^2 \sin(45^\circ) \cdot (0.80) \cdot (0.3491\text{ rad}) \cdot (0.4363\text{ rad}) = 0.77543Exact Volume V=∫θθ+dθ ⁣∫ϕϕ+dϕ ⁣∫ρρ+dρ ⁣r2sin⁡ϕ dr dϕ dθ=(30.80)3−(3.00)33[cos⁡(45∘)−cos⁡(65∘)](0.4363)=1.15326\text{Exact Volume } V = \int_{\theta}^{\theta+d\theta}\!\int_{\phi}^{\phi+d\phi}\!\int_{\rho}^{\rho+d\rho}\! r^2\sin\phi\, dr\,d\phi\,d\theta = \frac{(30.80)^3 - (3.00)^3}{3} \Big[ \cos(45^\circ) - \cos(65^\circ) \Big] (0.4363) = 1.15326

Special Plane Curves - Theory & Concepts - Analytic Geometry Spirograph

Analysis of advanced planar curves including cycloids, epicycloids, hypocycloids, and lemniscates.

Spirograph & Cycloid Simulator

Trace epicycloid and hypocycloid curves by rolling a circle along the outside or inside of a fixed circle

Fixed Radius R3.0
Rolling Radius r1.0
Angle: 0°
{x(θ)=(R−r)cos⁡θ+rcos⁡(R−rrθ)y(θ)=(R−r)sin⁡θ−rsin⁡(R−rrθ)\begin{cases} x(\theta) = (R-r)\cos\theta + r\cos\left(\frac{R-r}{r}\theta\right) \\ y(\theta) = (R-r)\sin\theta - r\sin\left(\frac{R-r}{r}\theta\right) \end{cases}

Special Plane Curves - Theory & Concepts - Analytic Geometry Catenary Cable

Analysis of advanced planar curves including cycloids, epicycloids, hypocycloids, and lemniscates.

Catenary vs Parabolic Cable Comparison

Span (Width S)10.0 m
Sag (Depth H)4.0 m
Cable Comparisons
Catenary Length:13.442 m
Parabola Length:13.337 m
Max Separation:0.147 m
Catenary Param a:3.646
Parabola Coeff k:0.1600

Catenary: Curve formed by a uniform hanging chain under its own weight. Governing equation: y=a(cosh⁡(x/a)−1)y = a(\cosh(x/a) - 1).

Parabola: Curve formed by a cable supporting a uniform horizontal load (like suspension bridge decks). Governing equation: y=kx2y = k x^2.

x = -S/2x = S/2CatenaryParabola
Mathematical Equations
Catenary: y=3.65(cosh⁡(x3.65)−1)\text{Catenary: } y = 3.65 \left( \cosh\left( \frac{x}{3.65} \right) - 1 \right)Parabola: y=0.1600x2\text{Parabola: } y = 0.1600 x^2