Artemis Academy · Fourier wordmark — ARTEMIS / Academy traced by 140 rotating vectors

Example 0 · The logoThis logo is drawn by the math in Example 1 — a sum of rotating vectors is a parametric curve. The letterforms are resampled into one parametric path, decomposed with a discrete Fourier transform, and redrawn by 140 rotating vectors, largest first; the tip of the chain leaves the gold ink trail that writes ARTEMIS over Academy. Watch it write, hold, fade, and loop — or press Replay.

ARTEMIS ACADEMY · An interactive lesson · 10 live demos · 8 themes · no pasting required

One knob makes a curve. Two knobs make a world.

An Artemis Academy lesson. Parametric modeling and multivariable calculus, in a nutshell — told through ten demos that run right here. Drag the points, turn the knobs, orbit the surfaces. Each one is also copy-paste ready for Desmos or GeoGebra, so you can rebuild it live in class. Use the theme bar above to switch the entire page — every card, control, table and demo — between Day, Night, Matrix, Matrix Blue, Matrix Light Blue, Manimx Day, Manimx Light Blue and Manimx Executive.

Parametric — the 10-second version

One knob t (think: time) drives x and y together. You don’t describe a shape — you describe a journey that traces it.

Multivariable — the other 10 seconds

Now there are 2+ knobs. A function f(x, y) is a landscape: its derivatives are slopes on that landscape, its integral is the volume under it.

01 · START HEREThe Circle That Isn’t a FunctionWhy y = f(x) isn’t enough — and motion appears 02Lissajous FiguresFrequency ratios become knots, stars & bows 03 · NEW · 2D → 3D BRIDGEThe 3D HelixA circle that climbs — springs, screws & DNA 04 · THE BIG ONEBézier CurvesThe actual math inside fonts, CAD & car bodies 05The CycloidA rolling wheel invents the fastest slide 06 · NEW DIMENSIONThe Mountainf(x, y), contour lines & the gradient 07Slicing the MountainPartial derivatives, demystified 08Torus & Möbius StripTwo knobs make a skin — one of them has one side 09 · NEW · MODELING FINALEFrenet-Swept PipeWhat a CAD kernel actually computes 10 · FINALEVolume + The ButterflyDouble integrals, then one formula of pure wow

▶ Start with Example 1 Tip: use ← → arrow keys to move between examples · every graph below is live — press Play, drag, slide.

The Circle That Isn’t a Function

Parametric basics. A circle fails the vertical line test, so y = f(x) can never draw one. Give an angle t control of both coordinates at once — and the shape gains something new: motion.

Level · Simple

Say this

Presenter line“Parametric is motion, not just shape. The slider s is time — watch the dot travel the circle instead of the circle just sitting there.”

Paste into Desmos 2D

a = 2 (a·cos(t), a·sin(t)) point: (a·cos(s), a·sin(s)) ← press Play on s

Open Desmos 2D ↗

x = a·cos t and y = a·sin t — one input, two outputs. That’s the whole trick.
Same idea draws ellipses, spirals and orbits: just change the two recipes.

Calculus Hookr(t) = (a cos t, a sin t) · r′(t) = (−a sin t, a cos t)
Velocity is always perpendicular to position: r · r′ = 0 for every t. Uniform circular motion, in one line — speed |r′| = a is constant.

radius a 2.0 angle s 0.80

x = a·cos(s) = —y = a·sin(s) = —speed is constant: the dot never hurries

Lissajous — Frequency Becomes Art

Let x wiggle p times while y wiggles q times. Change nothing but the ratio p : q and the same two sine waves weave knots, stars and bows.

Level · Simple

Say this

Presenter line“This is oscilloscopes, musical harmony and orbital resonance — one math. Integer ratios close the loop; messy ratios never quite repeat.”

Paste into Desmos 2D

p = 3 q = 4 d = 1.57 (sin(p·t), sin(q·t + d))

p : q = 1 : 1 with phase π/2 gives a circle — Example 1 was hiding inside this one.
The phase d twists the figure without changing its frequencies.

Calculus HookThe curve closes iff p/q ∈ ℚ. If p/q is irrational it never repeats — it comes arbitrarily close to every point in the square. Velocity is r′(t) = (p cos pt, q cos(qt+d)); it can vanish only when both cosines hit zero at the same t — for special (p, q, d) that is exactly where the figure develops a cusp or retraces itself (e.g. p = q = 1, d = 0 collapses to a line segment traced back and forth).

p 3 q 4 phase d 1.57

ratio p : q = 3 : 4closed curve — it retraces itself forever

The 3D Helix — The 2D → 3D Bridge

Take Example 1’s circle and let it climb while it turns. One parameter still drives everything, but the curve now lives in space: a spring, a screw thread, a DNA backbone, a spiral stair.

Level · Bridge to 3D

Say this

Presenter line“Nothing new was invented — we just gave the circle a third recipe, z = c·t. That is the whole jump from 2D curves to space curves.”

Paste into Desmos 3D / GeoGebra 3D

(R·cos t, R·sin t, c·t) t: 0 → 10π · R = radius · c = climb per radian

Open Desmos 3D ↗

R sets the coil radius; c sets the pitch — how fast it climbs per turn. Pitch = 2πc.
Project it flat (look down the axis) and you see Example 1’s circle again.

Calculus Hookr(t) = (R cos t, R sin t, c t) · r′(t) = (−R sin t, R cos t, c)
Speed |r′| = √(R² + c²) is constant — the tracer never speeds up. The helix is a geodesic on its cylinder (unroll the cylinder and it becomes a straight line) and the model of uniform motion on a screw. Curvature κ = R/(R²+c²), torsion τ = c/(R²+c²) — both constant.

radius R 1.00 climb c 0.22 ↖ Drag to orbit

speed |r′| = √(R²+c²) = — (constant)pitch = 2πc = —

Bézier Curve — This Is Parametric Modeling

Four points. One formula. Drag the orange control points and watch a perfectly smooth curve follow — this exact math draws every letter you’re reading, every car body, wing and CAD spline.

Level · Core idea

Say this

Presenter line“You never move the curve. You move two handles, and the math guarantees smoothness. Every modern design tool is this idea, scaled up.”

Paste into Desmos 2D — then drag A, B, C, D

A=(-4,-2) B=(-2,3) C=(3,3) D=(4,-1) (1-t)³A + 3(1-t)²t·B + 3(1-t)t²·C + t³D with 0 ≤ t ≤ 1

The weights (1−t)³, 3(1−t)²t, 3(1−t)t², t³ always sum to 1 — the point is a moving weighted average.
The faint lines show de Casteljau’s construction: averages of averages of averages.
Chain Béziers end-to-end and you get fonts, logos, animation paths, Revit splines.

Calculus HookThe Bernstein weights are a partition of unity: (1−t)³+3(1−t)²t+3(1−t)t²+t³ = ((1−t)+t)³ = 1. The derivative is itself a Bézier curve of the differences: B′(t) = 3[(1−t)²(B−A)+2(1−t)t(C−B)+t²(D−C)] — so endpoints are tangent to AB and CD.

t 0.35 ↖ Drag A · B · C · D directly on the graph

point at t: —curve always stays inside the control points’ hull

Cycloid — The Rolling Wheel & the Fastest Slide

Fix a dot to the rim of a rolling wheel. It traces neither circle nor line — it invents a new curve. Flip that curve over and it becomes the brachistochrone: the fastest possible slide, beating a straight line while being longer.

Level · Medium

Say this

Presenter line“At the bottom of each arch the dot is momentarily stopped — that’s why the cusps are sharp. Galileo, Bernoulli and Newton all fought over this curve.”

Paste into Desmos 2D

(t − sin(t), 1 − cos(t)) rolling circle: center (s, 1), radius 1

One arch spans exactly 2πr wide and 2r tall — and its arc length is exactly 8r, a clean whole number from a rolling circle.
In the race, both beads drop the same height. The cycloid dives steeply early, banks speed, and wins.

Calculus HookSpeed |r′(t)| = 2|sin(t/2)| — exactly 0 at each cusp, maximum at the top of the arch. Arc length of one arch is ∫₀²π 2|sin(t/2)| dt = 8r. The inverted cycloid also solves the calculus-of-variations problem: it minimises descent time (brachistochrone) and is tautochrone — descent time is independent of starting point.

roll s 2.00 Watch the red dot: it stops dead at each cusp.

tracing point: —one arch = 2π wide, 2 tall (r = 1)

The Mountain — Your First Multivariable Function

Two inputs, one output: the graph stops being a line and becomes a landscape. Slice it with a flat plane at height k and the intersection is a contour line. Drag to orbit; the arrow is the gradient — straight uphill, always perpendicular to the contour.

Level · New dimension

Say this

Presenter line“A topographic map is just this surface, sliced. And that’s why water flows across contour lines, never along them — it follows the gradient, downhill.”

Paste into GeoGebra 3D

f(x,y) = 3·exp(−x² − y²) + 2·exp(−(x−2)² − (y−2)²) Plane: z = k (slider k, 0 to 3)

Open GeoGebra 3D ↗Open Desmos 3D ↗

Level curve: all (x, y) where f(x, y) = k — one flat slice of the mountain.
Gradient ∇f at the marker: direction of steepest climb, length = steepness.
Move the marker — near a peak the arrow shrinks to nothing. Peaks are where ∇f = 0.

Calculus Hook∇f ⊥ level sets. Proof in one line: along a level curve r(t) with f(r(t)) = k, differentiating gives ∇f · r′ = 0. So ∇f is normal to the contour, points in the direction of steepest ascent, and |∇f| is that steepest rate. The directional derivative is D_uf = ∇f · u.

slice k 1.20 marker x 0.8 marker y 0.6 ↖ Drag the 3D view to orbit · bottom: contour map, top-down

3D Rendering Shaded light · Occlusion on — hidden surfaces removed (solid look)

Occlusion + light modes adopted from the Desmos 3D graphers in the Reference Library below.

height f = —∇f = —|∇f| = —

Slicing the Mountain — Partial Derivatives

A partial derivative is not mysterious. Freeze one variable, slice the mountain, and take an ordinary 1D slope on that slice. Do it in x and in y, and you know the full tilt of the surface at a point.

Level · Medium

Say this

Presenter line“∂f/∂x answers one question: if I walk purely east, how steep is it right here? ∂f/∂y asks the same walking north. Two honest 1D questions.”

Paste into GeoGebra 3D (continue from #6)

g(x) = f(x, b) ← slice, y frozen at b h(y) = f(a, y) ← slice, x frozen at a ∂f/∂x = slope of g at a ∂f/∂y = slope of h at b

The teal curve is the east–west slice; the orange curve is north–south. Their tangent lines are drawn at the marker.
Together the two slopes define the tangent plane — the flat sheet that just kisses the surface.
Gradient check: ∇f = ( ∂f/∂x , ∂f/∂y ) — exactly the arrow from Example 6.

Calculus HookThe two partials are the gradient: ∇f = (f_x, f_y), and they define the tangent plane z = f(a,b) + f_x(a,b)(x−a) + f_y(a,b)(y−b) — the best linear approximation, with error o(√((x−a)²+(y−b)²)) when f is differentiable.

Go deeper · Monkey Saddlez = x³ − 3xy², ∇f = (3x²−3y², −6xy), so ∇f(0,0) = (0,0) — yet the Hessian there is the zero matrix (degenerate). The surface has three valleys and three ridges (room for a monkey’s two legs and tail). The second-derivative test cannot classify this critical point — a failure mode with no 1D analogue.

point a (x) 0.8 point b (y) 0.6 ↖ Drag the 3D view to orbit · bottom: the two slices, flattened

3D Rendering Shaded light · Occlusion on — hidden surfaces removed (solid look)

Occlusion + light modes adopted from the Desmos 3D graphers in the Reference Library below.

∂f/∂x = —∂f/∂y = —steepest slope |∇f| = —

Torus & Möbius — Parametric Surfaces

A curve needed one parameter. A skin needs two — u and v, like latitude and longitude, or the warp and weft of fabric. Every 3D model, game character and architectural shell is stored exactly this way.

Level · Advanced

Say this

Presenter line“Switch to the Möbius strip and follow the travelling dot. It goes all the way around u… and arrives back upside-down. One trip proves the strip has only one side.”

Paste into Desmos 3D / GeoGebra 3D

TORUS: ((2+0.7cos v)cos u, (2+0.7cos v)sin u, 0.7 sin v) MÖBIUS: ((1+ v/2·cos(u/2))cos u, (1+ v/2·cos(u/2))sin u, v/2·sin(u/2)) u: 0→2π · torus v: 0→2π · möbius v: −1→1

Open Desmos 3D ↗

Torus: u goes around the donut, v goes around the tube. Two circles, multiplied.
Möbius: the half-twist cos(u/2) is the entire trick — one half-turn baked into the formula.
Game engines call this grid a mesh; CAD calls it a parametric surface. Same object.

Calculus HookSurface area is ∬ |r_u × r_v| du dv. For the torus |r_u × r_v| = r(R + r cos v), so area = ∫₀²π∫₀²π r(R+r cos v) du dv = 4π²Rr and volume = 2π²Rr². The Möbius strip is non-orientable: no continuous unit normal exists over it, so flux ∬ F·n dS is not defined until you cut it.

Go deeper · Helicoid & Torus Knot (2,3)Helicoid: r(u,v) = (u cos v, u sin v, c v) — a ruled minimal surface (mean curvature H = 0), the soap-film shape on a helical frame. Torus knot (2,3): compose the torus with (u,v) = (2t, 3t) to get a single curve that winds twice around the hole and three times through it — a surface becoming a curve by composition.

tube r 0.70 ↖ Drag to orbit

3D Rendering Shaded light · Occlusion on — far side hidden (solid look)

Occlusion + light modes adopted from the Desmos 3D graphers in the Reference Library below.

travelling dot: u = 0.00 — watch it come back flipped on the Möbius

Frenet-Swept Pipe — What CAD Actually Computes

The modeling finale. Take the helix from Example 3 as a spine, attach a moving frame to it, and sweep a small circle along it. The result is a solid pipe — a duct run, a thread, a handrail.

Level · Modeling finale

Say this

Presenter line“Curve + frame + profile = solid. This is not a picture of a pipe — it is the same boundary representation a CAD kernel builds when you route a duct or a pipe in Revit.”

Paste into GeoGebra 3D

r(t) = (R cos t, R sin t, c t) T = r′/|r′| · N = T′/|T′| · B = T × N P(t,θ) = r(t) + ρ(cos θ·N + sin θ·B) t: 0 → 4π · θ: 0 → 2π · ρ = pipe radius

Open GeoGebra 3D ↗

The bright ring is the profile circle at the travelling point — watch it stay perpendicular to the spine (that’s the Frenet frame doing its job).
Change ρ and you change wall thickness of the sweep, not the path.

Calculus HookFor the helix, with s = √(R²+c²): T = (−R sin t, R cos t, c)/s, N = (−cos t, −sin t, 0), B = T × N. The swept surface is P(t,θ) = r(t) + ρ(cos θ N + sin θ B) — a parametric surface whose two knobs are along (t) and around (θ). This curve + frame + profile construction is the B-rep a CAD kernel computes — behind every pipe, duct, thread and digital twin.

spine R 1.50 pipe ρ 0.24 climb c 0.22 ↖ Drag to orbit

3D Rendering Shaded light · Occlusion on — back wall hidden (solid pipe)

Occlusion + light modes adopted from the Desmos 3D graphers in the Reference Library below.

sweep point t = —profile stays ⊥ to spine via T · N · B frame

Finale — Volume Under a Surface, Then the Butterfly

First the payoff of multivariable calculus: fill a dome with thin columns — base dx·dy, height f(x, y) — and add them all. That sum is the double integral. Then, one parameter and one line of math… a butterfly.

Level · Finale

Say this

Presenter line“A 1D integral adds rectangles to get area. A double integral adds columns to get volume. Same idea, one dimension up — refine the grid and watch the estimate lock onto the exact answer.”

Paste into GeoGebra 3D / Desmos

VOLUME: f(x,y) = 4 − x² − y² over x² + y² ≤ 1 ∬ f dA = 7π/2 ≈ 10.996 BUTTERFLY (Desmos 2D, 0 ≤ t ≤ 24π): r = e^{sin t} − 2cos(4t) + sin⁵((2t−π)/24) (r·sin t, r·cos t)

Grid slider = the limit process, live: more, thinner columns → sum → exact value.
The butterfly (Temple Fay’s curve) is compression: a one-line rule generating a shape no one could plot point-by-point.

Calculus HookThe column sum is a Riemann sum: Σ f(x_i,y_j) ΔA → ∬_D f dA as n → ∞ (Fubini lets you compute it as iterated 1D integrals). Here ∬ (4−x²−y²) dA = 7π/2 over the unit disk — in polar coordinates the Jacobian r dr dθ does the heavy lifting. For the butterfly, arc length has no elementary closed form, yet the parametric integral still converges.

grid n × n 9 ↖ Drag to orbit the dome

3D Rendering Shaded light · Occlusion on — columns behind the dome hidden (solid look)

Occlusion + light modes adopted from the Desmos 3D graphers in the Reference Library below.

column sum Σ f·ΔA = —exact 7π/2 = 10.996error = —

The memory hook — end on this

If they remember five symbols, they remember the subject.

1 parameter →a curve, with motion built in

f(x, y) →a surface — a landscape over two inputs

2 parameters →a skin: torus, Möbius, every 3D model

∂ →slope of the surface, one frozen direction at a time

∬ →volume under the surface, column by column

Parametric equations are compression: simple rules, turned by one or two knobs, generating shapes — circles, slides, mountains, donuts, butterflies — that no static formula could draw. That is the whole subject. Everything else is turning the knobs with more care.

Keep exploring · the presenter's picks

Surface & Solid Modeling — Reference Library

Two Desmos 3D graphers are the stars here — both graph explicit functions z = f(x, y) inside the regular Desmos calculator, and both are loved for the same two features: occlusion (hidden-surface removal, so a surface reads as a solid) and light modes (shading styles that reveal shape). Those two features are now built into this page's own 3D demos above — toggle Shaded / Flat / Wireframe and Occlusion On / Off on the Mountain, Surfaces, Swept Pipe and Volume demos — and the originals are one click away below, alongside the platforms every paste-in formula on this page runs on.

Reference	What it shows	Modeling type	Use it for	Link
3D Occlusion Light modes Grapher — explicit functionFull title: “3D Occlusion Light modes Grapher explicit function — Graficador 3D con ocultación modos de luz función explícita” · Desmos calculator graph	A 3D grapher for surfaces z = f(x, y) built inside the 2D Desmos calculator, with occlusion (hidden-surface removal) and switchable light modes for shading surfaces and solids.	Surface · Solid shading	The occlusion + light-mode look adopted on this page — compare its solid, shaded surfaces with the Mountain and Volume demos above, then change the light mode and watch the shape reading change.	Open in Desmos ↗desmos.com/calculator/aqvl7rvzo6
3D Light Grapher — explicit functionFull title: “3D Light Grapher explicit function — Graficador 3D luz función explícita” · Desmos calculator graph	A companion 3D grapher for z = f(x, y) centred on lighting and shading of the surface — the light-mode companion to the occlusion grapher above.	Surface · Light modes	Studying how directional light and shading modes make curvature, peaks and valleys legible on a surface or solid — the same idea as the Shaded / Flat / Wireframe toggle on this page's demos.	Open in Desmos ↗desmos.com/calculator/aftgpjekqs
Desmos Graphing CalculatorThe 2D platform used throughout this presentation	The standard Desmos 2D calculator — where the Circle, Lissajous, Bézier, Cycloid and Butterfly formulas on this page paste straight in.	2D curves	Rebuilding Examples 1, 2, 4, 5 and the Butterfly live in class, with sliders for every parameter.	Open Desmos ↗desmos.com/calculator
Desmos 3DDesmos's native 3D grapher	Native 3D graphing of surfaces and space curves — the Helix, Torus and Möbius formulas on this page run here directly.	3D surfaces	Examples 3 and 8 live in 3D, plus quick z = f(x, y) surfaces to compare against the two graphers above.	Open Desmos 3D ↗desmos.com/3d
GeoGebra 3DThe 3D platform for the multivariable examples	GeoGebra's 3D calculator — where the Mountain, Partials, Swept Pipe and Volume formulas on this page are written for.	Surfaces · Solids	Examples 6, 7, 9 and 10: surfaces with slicing planes, partial-derivative slices, sweeps and double-integral volume.	Open GeoGebra 3D ↗geogebra.org/3d

Note: the second Desmos grapher (…/aftgpjekqs) was listed twice by the presenter — once with the surface/solid picks and again as “this 3D” — so it appears once here; both mentions point to the same graph. Titles above are the graphs' own Desmos page titles, verified 2026-11-02.