# From Curves to Surfaces > _The name of a surface tells you what you see when you slice it._ --- ## Part I — The Three Conics All three curves come from one object: a double cone, two ice cream cones touching tip to tip. Slice it with a flat plane. The angle of the cut determines which curve appears. |Curve|Etymology|The cut| |---|---|---| |**Ellipse**|_elleipsis_ — falling short|Shallower than the slope → closed loop| |**Parabola**|_para bolē_ — thrown beside|Exactly parallel to slope → open, one direction| |**Hyperbola**|_hyper bolē_ — thrown beyond|Steeper than slope, cuts both cones → two branches| The progression: **closed → open → split apart**. ### Ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ Both squared terms are positive, summing to 1. Bounded, closed, finite. <svg viewBox="-120 -90 240 170" xmlns="http://www.w3.org/2000/svg" style="max-width:300px;display:block"> <line x1="-110.0" y1="0.0" x2="110.0" y2="0.0" stroke="#aaa" stroke-width="1" opacity="0.7"/> <line x1="0.0" y1="-90.0" x2="0.0" y2="90.0" stroke="#aaa" stroke-width="1" opacity="0.7"/> <ellipse cx="0" cy="0" rx="90" ry="55" fill="none" stroke="#5B9BD5" stroke-width="2.5"/> <text x="95.0" y="5.0" fill="#aaa" font-size="11" font-family="serif" text-anchor="start">x</text> <text x="5.0" y="-72.0" fill="#aaa" font-size="11" font-family="serif" text-anchor="start">y</text> <text x="-108.0" y="-60.0" fill="#5B9BD5" font-size="11" font-family="serif" text-anchor="start">x²/a² + y²/b² = 1</text> <text x="-28.0" y="8.0" fill="#5B9BD5" font-size="10" font-family="serif" text-anchor="middle">a</text> <text x="-106.0" y="30.0" fill="#5B9BD5" font-size="10" font-family="serif" text-anchor="middle">b</text> <line x1="0.0" y1="0.0" x2="90.0" y2="0.0" stroke="#5B9BD5" stroke-width="1.2" opacity="0.6" stroke-dasharray="4,3"/> <line x1="0.0" y1="0.0" x2="0.0" y2="-55.0" stroke="#5B9BD5" stroke-width="1.2" opacity="0.6" stroke-dasharray="4,3"/> </svg> ### Parabola $y = \frac{x^2}{a^2}$ One variable is squared; the other is **not**. The unsquared variable is the axis it opens along. No second branch — it sits at the tipping point between closed and split. <svg viewBox="-120 -90 240 115" xmlns="http://www.w3.org/2000/svg" style="max-width:300px;display:block"> <line x1="-110.0" y1="0.0" x2="110.0" y2="0.0" stroke="#aaa" stroke-width="1" opacity="0.7"/> <line x1="0.0" y1="-85.0" x2="0.0" y2="85.0" stroke="#aaa" stroke-width="1" opacity="0.7"/> <path d="M -64.0,-74.5 L -60.0,-65.5 L -56.0,-57.0 L -52.0,-49.2 L -48.0,-41.9 L -44.0,-35.2 L -40.0,-29.1 L -36.0,-23.6 L -32.0,-18.6 L -28.0,-14.3 L -24.0,-10.5 L -20.0,-7.3 L -16.0,-4.7 L -12.0,-2.6 L -8.0,-1.2 L -4.0,-0.3 L 0.0,-0.0 L 4.0,-0.3 L 8.0,-1.2 L 12.0,-2.6 L 16.0,-4.7 L 20.0,-7.3 L 24.0,-10.5 L 28.0,-14.3 L 32.0,-18.6 L 36.0,-23.6 L 40.0,-29.1 L 44.0,-35.2 L 48.0,-41.9 L 52.0,-49.2 L 56.0,-57.0 L 60.0,-65.5 L 64.0,-74.5" fill="none" stroke="#E8A838" stroke-width="2.5" opacity="0.85"/> <text x="95.0" y="5.0" fill="#aaa" font-size="11" font-family="serif" text-anchor="start">x</text> <text x="5.0" y="-78.0" fill="#aaa" font-size="11" font-family="serif" text-anchor="start">y</text> <text x="-108.0" y="-50.0" fill="#E8A838" font-size="11" font-family="serif" text-anchor="start">y = x²/a²</text> <text x="8.0" y="10.0" fill="#aaa" font-size="9" font-family="serif" text-anchor="start">vertex</text> </svg> ### Hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ One sign flips negative. That single change tears the curve into two branches. Dashed lines are asymptotes. <svg viewBox="-120 -100 240 200" xmlns="http://www.w3.org/2000/svg" style="max-width:300px;display:block"> <line x1="-110.0" y1="0.0" x2="110.0" y2="0.0" stroke="#aaa" stroke-width="1" opacity="0.7"/> <line x1="0.0" y1="-90.0" x2="0.0" y2="90.0" stroke="#aaa" stroke-width="1" opacity="0.7"/> <line x1="-108.0" y1="87.2" x2="108.0" y2="-87.2" stroke="#666" stroke-width="1" opacity="0.5" stroke-dasharray="5,4"/> <line x1="-108.0" y1="-87.2" x2="108.0" y2="87.2" stroke="#666" stroke-width="1" opacity="0.5" stroke-dasharray="5,4"/> <path d="M 115.9,83.7 L 104.3,73.0 L 94.2,63.4 L 85.4,54.7 L 77.9,46.8 L 71.5,39.6 L 66.1,32.9 L 61.6,26.7 L 58.1,20.9 L 55.4,15.4 L 53.5,10.2 L 52.4,5.1 L 52.0,-0.0 L 52.4,-5.1 L 53.5,-10.2 L 55.4,-15.4 L 58.1,-20.9 L 61.6,-26.7 L 66.1,-32.9 L 71.5,-39.6 L 77.9,-46.8 L 85.4,-54.7 L 94.2,-63.4 L 104.3,-73.0 L 115.9,-83.7" fill="none" stroke="#9B59B6" stroke-width="2.5" opacity="0.85"/> <path d="M -115.9,83.7 L -104.3,73.0 L -94.2,63.4 L -85.4,54.7 L -77.9,46.8 L -71.5,39.6 L -66.1,32.9 L -61.6,26.7 L -58.1,20.9 L -55.4,15.4 L -53.5,10.2 L -52.4,5.1 L -52.0,-0.0 L -52.4,-5.1 L -53.5,-10.2 L -55.4,-15.4 L -58.1,-20.9 L -61.6,-26.7 L -66.1,-32.9 L -71.5,-39.6 L -77.9,-46.8 L -85.4,-54.7 L -94.2,-63.4 L -104.3,-73.0 L -115.9,-83.7" fill="none" stroke="#9B59B6" stroke-width="2.5" opacity="0.85"/> <text x="95.0" y="15.0" fill="#aaa" font-size="11" font-family="serif" text-anchor="start">x</text> <text x="5.0" y="-82.0" fill="#aaa" font-size="11" font-family="serif" text-anchor="start">y</text> <text x="-108.0" y="-68.0" fill="#9B59B6" font-size="11" font-family="serif" text-anchor="start">x²/a² − y²/b² = 1</text> </svg> --- ## Part II — The Sign Grammar Before naming any surface, run through three questions in order. **Question 1: Is any variable missing entirely?** If yes — that variable is the axis. The surface is a **cylinder**: the 2D curve in the remaining two variables, extruded infinitely. Nothing more. **Question 2: Is any variable unsquared (linear)?** If yes — that variable is the output axis, and the surface is a **paraboloid**. If the remaining squared terms share the same sign, it is an **elliptic paraboloid** (bowl). If they differ in sign, it is a **hyperbolic paraboloid** (saddle). **Question 3: All variables are squared — count the minus signs.** - Zero minus signs, right-hand side = 1 → **Ellipsoid** - Zero minus signs, right-hand side = 0 → **Elliptic cone** (boundary case) - One minus sign, right-hand side = 1 → **Hyperboloid of one sheet** - One minus sign (or two), right-hand side = −1 → **Hyperboloid of two sheets** > **Memory hook for hyperboloids:** count the minus signs on the left. One minus → one sheet. Flip the right-hand side to −1 (equivalently, two minus signs on one side) → two sheets. Complete the square first whenever the surface is shifted from the origin — that clears the way for these three questions to work cleanly. --- ## Part III — The Surfaces ### Ellipsoid $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$ Every cross-section in every direction is an ellipse. When $a = b = c$ it becomes a sphere. 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34.1,37.8 L 33.3,35.8 L 32.4,33.7 L 31.6,31.7 L 30.9,29.6 L 30.1,27.6 L 29.4,25.5 L 28.8,23.4 L 28.1,21.3 L 27.5,19.2 L 27.0,17.1 L 26.4,15.0 L 26.0,12.9 L 25.5,10.8 L 25.2,8.7 L 24.9,6.5 L 24.6,4.4 L 24.4,2.2 L 24.2,0.0 L 24.1,-2.2 L 24.1,-4.4 L 24.1,-6.6 L 24.2,-8.8 L 24.4,-11.0 L 24.6,-13.2 L 24.9,-15.5 L 25.2,-17.7 L 25.5,-20.0 L 26.0,-22.3 L 26.4,-24.6 L 27.0,-26.9 L 27.5,-29.2 L 28.1,-31.5 L 28.8,-33.8 L 29.4,-36.1 L 30.1,-38.4 L 30.9,-40.8 L 31.6,-43.1 L 32.4,-45.5 L 33.3,-47.8 L 34.1,-50.2 L 35.0,-52.5 L 35.8,-54.9 L 36.7,-57.2 L 37.7,-59.6 L 38.6,-62.0 L 39.5,-64.3 L 40.5,-66.7 L 41.5,-69.1 L 42.5,-71.5 L 43.5,-73.9" fill="none" stroke="#3DBFBF" stroke-width="1.3" opacity="0.7"/> <text x="-95.0" y="-92.0" fill="#3DBFBF" font-size="11" font-family="serif" text-anchor="start">x²/a² + y²/b² − z²/c² = 1</text> </svg> --- ### Hyperboloid of Two Sheets $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = -1$ Right-hand side flips to $-1$. The surface can no longer stay connected — **two separate bowls** floating apart from each other along the $z$-axis. 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33.9,-82.0 L 34.6,-83.3" fill="none" stroke="#E05C5C" stroke-width="1.3" opacity="0.7"/> <path d="M 5.4,44.1 L 7.7,44.8 L 9.5,45.6 L 11.0,46.4 L 12.4,47.3 L 13.7,48.1 L 14.9,49.0 L 16.0,49.9 L 17.1,50.8 L 18.1,51.7 L 19.1,52.7 L 20.0,53.6 L 21.0,54.5 L 21.9,55.4 L 22.8,56.4 L 23.6,57.3 L 24.5,58.3 L 25.3,59.2 L 26.1,60.2 L 27.0,61.1 L 27.8,62.1 L 28.6,63.0 L 29.3,64.0 L 30.1,65.0 L 30.9,65.9 L 31.6,66.9 L 32.4,67.8 L 33.1,68.8 L 33.9,69.8 L 34.6,70.7" fill="none" stroke="#E05C5C" stroke-width="1.3" opacity="0.7"/> <text x="-95.0" y="-102.0" fill="#E05C5C" font-size="11" font-family="serif" text-anchor="start">x²/a² + y²/b² − z²/c² = −1</text> </svg> --- ### Elliptic Cone $\frac{z^2}{c^2} = \frac{x^2}{a^2} + \frac{y^2}{b^2}$ Right-hand side is **zero** — the boundary case between one-sheet and two-sheet hyperboloids. Fix $z$: ellipses. Fix $x$ or $y$: **two lines crossing at the origin** (a degenerate hyperbola). 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19.3,18.5 L 17.4,16.7 L 15.4,14.8 L 13.5,13.0 L 11.6,11.1 L 9.6,9.3 L 7.7,7.4 L 5.8,5.6 L 3.9,3.7 L 1.9,1.9 L 0.0,-0.0 L 1.9,-2.5 L 3.9,-5.1 L 5.8,-7.6 L 7.7,-10.2 L 9.6,-12.7 L 11.6,-15.3 L 13.5,-17.8 L 15.4,-20.4 L 17.4,-22.9 L 19.3,-25.5 L 21.2,-28.0 L 23.1,-30.6 L 25.1,-33.1 L 27.0,-35.7 L 28.9,-38.2 L 30.9,-40.8 L 32.8,-43.3 L 34.7,-45.9 L 36.7,-48.4 L 38.6,-51.0 L 40.5,-53.5 L 42.4,-56.1 L 44.4,-58.6 L 46.3,-61.2 L 48.2,-63.7 L 50.2,-66.3 L 52.1,-68.8 L 54.0,-71.4 L 55.9,-73.9 L 57.9,-76.5" fill="none" stroke="#9B59B6" stroke-width="1.3" opacity="0.7"/> <circle cx="0.0" cy="0.0" r="3.5" fill="#9B59B6" opacity="0.9"/> <text x="-92.0" y="-88.0" fill="#9B59B6" font-size="11" font-family="serif" text-anchor="start">z²/c² = x²/a² + y²/b²</text> </svg> #### Elliptic Cone vs. a Real Cone A "real cone" — the kind you hold ice cream in — is just **one nappe**, finite, solid. The mathematical elliptic cone is fundamentally different in two ways. **First: it is a double surface.** Because $z$ appears squared, both $+z$ and $-z$ satisfy the equation equally. The upper nappe and lower nappe are mirror images, joined at the vertex. An ice cream cone suppresses the lower nappe entirely — it only shows you half the mathematical object. **Second: it is infinite.** The nappes extend forever. The ice cream cone is a finite, truncated approximation. The cone is also the parent of the conics themselves. Every conic section is literally a cross-section of this double cone — which is why the cone equation sits as a special case ($ ext{RHS} = 0$) among the hyperboloids. It is the pivot point between them. --- ## Part IV — The Cylinders A cylinder does **not** promote a curve into 3D. It extends the curve along one axis — unchanged, infinitely. The variable that is **missing** from the equation is the axis of extension. > **A -oid surface is a conic in conversation with all three dimensions. A cylinder is a conic ignoring one dimension entirely.** The cross-sections along the extension axis are all identical. Nothing changes as you travel along it. This is the defining character of a cylinder. --- ### Elliptic Cylinder $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $z$ is missing → extends along $z$. Every horizontal slice is the same ellipse. <svg viewBox="-105 -70 220 115" xmlns="http://www.w3.org/2000/svg" style="max-width:300px;display:block"> <line x1="-66.0" y1="0.0" x2="66.0" y2="0.0" stroke="#aaa" stroke-width="1" opacity="0.7"/> <line x1="36.3" y1="-18.5" x2="-36.3" y2="18.5" stroke="#aaa" stroke-width="1" opacity="0.7"/> <line x1="0.0" y1="44.0" x2="0.0" y2="-44.0" stroke="#aaa" stroke-width="1" opacity="0.7"/> <text x="72.4" y="5.0" fill="#aaa" font-size="11" font-family="serif" text-anchor="start">x</text> <text x="-35.7" y="24.7" fill="#aaa" font-size="11" font-family="serif" text-anchor="start">y</text> <text x="2.0" y="-48.4" fill="#aaa" font-size="11" font-family="serif" text-anchor="start">z</text> <path d="M 39.6,33.0 L 37.0,34.1 L 33.5,35.1 L 29.2,36.1 L 24.2,37.0 L 18.6,37.8 L 12.5,38.5 L 6.1,39.0 L -0.4,39.4 L -7.0,39.7 L -13.3,39.8 L -19.3,39.7 L -24.9,39.4 L -29.8,39.0 L -34.0,38.5 L -37.4,37.8 L -39.9,37.0 L -41.3,36.1 L -41.8,35.1 L -41.2,34.1 L -39.6,33.0 L -37.0,31.9 L -33.5,30.9 L -29.2,29.9 L -24.2,29.0 L -18.6,28.2 L -12.5,27.5 L -6.1,27.0 L 0.4,26.6 L 7.0,26.3 L 13.3,26.2 L 19.3,26.3 L 24.9,26.6 L 29.8,27.0 L 34.0,27.5 L 37.4,28.2 L 39.9,29.0 L 41.3,29.9 L 41.8,30.9 L 41.2,31.9 L 39.6,33.0 Z" fill="none" stroke="#5B9BD5" stroke-width="1.4" opacity="0.8"/> <path d="M 39.6,16.5 L 37.0,17.6 L 33.5,18.6 L 29.2,19.6 L 24.2,20.5 L 18.6,21.3 L 12.5,22.0 L 6.1,22.5 L -0.4,22.9 L -7.0,23.2 L -13.3,23.3 L -19.3,23.2 L -24.9,22.9 L -29.8,22.5 L -34.0,22.0 L -37.4,21.3 L -39.9,20.5 L -41.3,19.6 L -41.8,18.6 L -41.2,17.6 L -39.6,16.5 L -37.0,15.4 L -33.5,14.4 L -29.2,13.4 L -24.2,12.5 L -18.6,11.7 L -12.5,11.0 L -6.1,10.5 L 0.4,10.1 L 7.0,9.8 L 13.3,9.7 L 19.3,9.8 L 24.9,10.1 L 29.8,10.5 L 34.0,11.0 L 37.4,11.7 L 39.9,12.5 L 41.3,13.4 L 41.8,14.4 L 41.2,15.4 L 39.6,16.5 Z" fill="none" stroke="#5B9BD5" stroke-width="1.4" opacity="0.8"/> <path d="M 39.6,0.0 L 37.0,1.1 L 33.5,2.1 L 29.2,3.1 L 24.2,4.0 L 18.6,4.8 L 12.5,5.5 L 6.1,6.0 L -0.4,6.4 L -7.0,6.7 L -13.3,6.8 L -19.3,6.7 L -24.9,6.4 L -29.8,6.0 L -34.0,5.5 L -37.4,4.8 L -39.9,4.0 L -41.3,3.1 L -41.8,2.1 L -41.2,1.1 L -39.6,0.0 L -37.0,-1.1 L -33.5,-2.1 L -29.2,-3.1 L -24.2,-4.0 L -18.6,-4.8 L -12.5,-5.5 L -6.1,-6.0 L 0.4,-6.4 L 7.0,-6.7 L 13.3,-6.8 L 19.3,-6.7 L 24.9,-6.4 L 29.8,-6.0 L 34.0,-5.5 L 37.4,-4.8 L 39.9,-4.0 L 41.3,-3.1 L 41.8,-2.1 L 41.2,-1.1 L 39.6,-0.0 Z" fill="none" stroke="#5B9BD5" stroke-width="1.4" opacity="0.8"/> <path d="M 39.6,-16.5 L 37.0,-15.4 L 33.5,-14.4 L 29.2,-13.4 L 24.2,-12.5 L 18.6,-11.7 L 12.5,-11.0 L 6.1,-10.5 L -0.4,-10.1 L -7.0,-9.8 L -13.3,-9.7 L -19.3,-9.8 L -24.9,-10.1 L -29.8,-10.5 L -34.0,-11.0 L -37.4,-11.7 L -39.9,-12.5 L -41.3,-13.4 L -41.8,-14.4 L -41.2,-15.4 L -39.6,-16.5 L -37.0,-17.6 L -33.5,-18.6 L -29.2,-19.6 L -24.2,-20.5 L -18.6,-21.3 L -12.5,-22.0 L -6.1,-22.5 L 0.4,-22.9 L 7.0,-23.2 L 13.3,-23.3 L 19.3,-23.2 L 24.9,-22.9 L 29.8,-22.5 L 34.0,-22.0 L 37.4,-21.3 L 39.9,-20.5 L 41.3,-19.6 L 41.8,-18.6 L 41.2,-17.6 L 39.6,-16.5 Z" fill="none" stroke="#5B9BD5" stroke-width="1.4" opacity="0.8"/> <path d="M 39.6,-33.0 L 37.0,-31.9 L 33.5,-30.9 L 29.2,-29.9 L 24.2,-29.0 L 18.6,-28.2 L 12.5,-27.5 L 6.1,-27.0 L -0.4,-26.6 L -7.0,-26.3 L -13.3,-26.2 L -19.3,-26.3 L -24.9,-26.6 L -29.8,-27.0 L -34.0,-27.5 L -37.4,-28.2 L -39.9,-29.0 L -41.3,-29.9 L -41.8,-30.9 L -41.2,-31.9 L -39.6,-33.0 L -37.0,-34.1 L -33.5,-35.1 L -29.2,-36.1 L -24.2,-37.0 L -18.6,-37.8 L -12.5,-38.5 L -6.1,-39.0 L 0.4,-39.4 L 7.0,-39.7 L 13.3,-39.8 L 19.3,-39.7 L 24.9,-39.4 L 29.8,-39.0 L 34.0,-38.5 L 37.4,-37.8 L 39.9,-37.0 L 41.3,-36.1 L 41.8,-35.1 L 41.2,-34.1 L 39.6,-33.0 Z" fill="none" stroke="#5B9BD5" stroke-width="1.4" opacity="0.8"/> <path d="M 39.6,33.0 L 39.6,-33.0" fill="none" stroke="#5B9BD5" stroke-width="1.1" opacity="0.6"/> <path d="M 18.6,37.8 L 18.6,-28.2" fill="none" stroke="#5B9BD5" stroke-width="1.1" opacity="0.6"/> <path d="M -13.3,39.8 L -13.3,-26.2" fill="none" stroke="#5B9BD5" stroke-width="1.1" opacity="0.6"/> <path d="M -37.4,37.8 L -37.4,-28.2" fill="none" stroke="#5B9BD5" stroke-width="1.1" opacity="0.6"/> <path d="M -39.6,33.0 L -39.6,-33.0" fill="none" stroke="#5B9BD5" stroke-width="1.1" opacity="0.6"/> <path d="M -18.6,28.2 L -18.6,-37.8" fill="none" stroke="#5B9BD5" stroke-width="1.1" opacity="0.6"/> <path d="M 13.3,26.2 L 13.3,-39.8" fill="none" stroke="#5B9BD5" stroke-width="1.1" opacity="0.6"/> <path d="M 37.4,28.2 L 37.4,-37.8" fill="none" stroke="#5B9BD5" stroke-width="1.1" opacity="0.6"/> <text x="-95.0" y="-58.0" fill="#5B9BD5" font-size="11" font-family="serif" text-anchor="start">x²/a² + y²/b² = 1 (∀z)</text> </svg> --- ### Parabolic Cylinder $x^2 = 4py$ $z$ is missing → extends along $z$. The parabola in the $xy$-plane, cloned infinitely upward and downward. Contrast with the paraboloid: the paraboloid's elliptical cross-sections grow as you rise; this cylinder's parabolic cross-sections never change. <svg viewBox="-100 -70 215 120" xmlns="http://www.w3.org/2000/svg" style="max-width:300px;display:block"> <line x1="-66.0" y1="0.0" x2="66.0" y2="0.0" stroke="#aaa" stroke-width="1" opacity="0.7"/> <line x1="36.3" y1="-18.5" x2="-36.3" y2="18.5" stroke="#aaa" stroke-width="1" opacity="0.7"/> <line x1="0.0" y1="44.0" x2="0.0" y2="-44.0" stroke="#aaa" stroke-width="1" opacity="0.7"/> <text x="72.4" y="5.0" fill="#aaa" font-size="11" font-family="serif" text-anchor="start">x</text> <text x="-35.7" y="24.7" fill="#aaa" font-size="11" font-family="serif" text-anchor="start">y</text> <text x="2.0" y="-48.4" fill="#aaa" font-size="11" font-family="serif" text-anchor="start">z</text> <path d="M -84.5,46.9 L -80.2,45.8 L -76.0,44.8 L -71.9,43.9 L -67.9,42.9 L -64.0,42.1 L -60.1,41.2 L -56.4,40.4 L -52.7,39.7 L -49.1,38.9 L -45.5,38.3 L -42.1,37.6 L 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-20.2,-31.7 L -17.4,-32.0 L -14.7,-32.3 L -12.0,-32.5 L -9.4,-32.7 L -7.0,-32.8 L -4.6,-32.9 L -2.2,-33.0 L 0.0,-33.0 L 2.2,-33.0 L 4.2,-32.9 L 6.2,-32.8 L 8.2,-32.7 L 10.0,-32.5 L 11.7,-32.3 L 13.4,-32.0 L 15.0,-31.7 L 16.5,-31.3 L 18.0,-30.9 L 19.3,-30.5 L 20.6,-30.0 L 21.8,-29.5 L 22.9,-29.0 L 23.9,-28.4 L 24.9,-27.7 L 25.7,-27.1 L 26.5,-26.3 L 27.2,-25.6 L 27.9,-24.8 L 28.4,-23.9 L 28.9,-23.1 L 29.3,-22.1 L 29.6,-21.2 L 29.8,-20.2 L 29.9,-19.1" fill="none" stroke="#E8A838" stroke-width="1.4" opacity="0.8"/> <path d="M -80.2,45.8 L -80.2,-20.2" fill="none" stroke="#E8A838" stroke-width="1.1" opacity="0.6"/> <path d="M -42.1,37.6 L -42.1,-28.4" fill="none" stroke="#E8A838" stroke-width="1.1" opacity="0.6"/> <path d="M -12.0,33.5 L -12.0,-32.5" fill="none" stroke="#E8A838" stroke-width="1.1" opacity="0.6"/> <path d="M 10.0,33.5 L 10.0,-32.5" fill="none" stroke="#E8A838" stroke-width="1.1" opacity="0.6"/> <path d="M 23.9,37.6 L 23.9,-28.4" fill="none" stroke="#E8A838" stroke-width="1.1" opacity="0.6"/> <path d="M 29.8,45.8 L 29.8,-20.2" fill="none" stroke="#E8A838" stroke-width="1.1" opacity="0.6"/> <text x="-92.0" y="-58.0" fill="#E8A838" font-size="11" font-family="serif" text-anchor="start">x² = 4py (∀z)</text> </svg> --- ### Hyperbolic Cylinder $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ $z$ is missing → extends along $z$. Two separate sheets of the hyperbola, each extruded infinitely. The sign grammar of the hyperbola ($+$ then $-$) is fully present — just swept along $z$ without modification. <svg viewBox="-105 -70 220 120" xmlns="http://www.w3.org/2000/svg" style="max-width:300px;display:block"> <line x1="-66.0" y1="0.0" x2="66.0" y2="0.0" stroke="#aaa" stroke-width="1" opacity="0.7"/> <line x1="36.3" y1="-18.5" x2="-36.3" y2="18.5" stroke="#aaa" stroke-width="1" opacity="0.7"/> <line x1="0.0" y1="44.0" x2="0.0" y2="-44.0" stroke="#aaa" stroke-width="1" opacity="0.7"/> <text x="72.4" y="5.0" fill="#aaa" font-size="11" font-family="serif" text-anchor="start">x</text> <text x="-35.7" y="24.7" fill="#aaa" font-size="11" font-family="serif" text-anchor="start">y</text> <text x="2.0" y="-48.4" fill="#aaa" font-size="11" font-family="serif" text-anchor="start">z</text> <path d="M 79.8,21.3 L 72.6,22.5 L 66.1,23.7 L 60.2,24.8 L 55.0,25.8 L 50.3,26.7 L 46.1,27.5 L 42.3,28.3 L 39.0,29.1 L 36.1,29.8 L 33.5,30.5 L 31.3,31.1 L 29.4,31.8 L 27.7,32.4 L 26.4,33.0 L 25.3,33.6 L 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stroke="#5BAD7F" stroke-width="1.1" opacity="0.55"/> <path d="M -24.6,27.5 L -24.6,-38.5" fill="none" stroke="#5BAD7F" stroke-width="1.1" opacity="0.55"/> <path d="M -23.6,30.5 L -23.6,-35.5" fill="none" stroke="#5BAD7F" stroke-width="1.1" opacity="0.55"/> <path d="M -26.4,33.0 L -26.4,-33.0" fill="none" stroke="#5BAD7F" stroke-width="1.1" opacity="0.55"/> <path d="M -33.5,35.5 L -33.5,-30.5" fill="none" stroke="#5BAD7F" stroke-width="1.1" opacity="0.55"/> <path d="M -46.1,38.5 L -46.1,-27.5" fill="none" stroke="#5BAD7F" stroke-width="1.1" opacity="0.55"/> <path d="M -66.1,42.3 L -66.1,-23.7" fill="none" stroke="#5BAD7F" stroke-width="1.1" opacity="0.55"/> <text x="-95.0" y="-58.0" fill="#5BAD7F" font-size="11" font-family="serif" text-anchor="start">x²/a² − y²/b² = 1 (∀z)</text> </svg> --- ## Summary Table | Surface | Equation signature | Key cross-sections | | ------------------------- | -------------------------------------------- | -------------------------------------------------- | | **Ellipsoid** | $x^2 + y^2 + z^2 = 1$ (all $+$, RHS $= 1$) | Ellipses in all directions | | **Elliptic Paraboloid** | $z = x^2 + y^2$ (one linear) | Ellipses horizontal, parabolas vertical | | **Hyperbolic Paraboloid** | $z = x^2 - y^2$ (one linear, opposite signs) | Hyperbolas horizontal, opposing parabolas vertical | | **Hyperboloid 1 sheet** | $x^2 + y^2 - z^2 = 1$ (one $-$, RHS $= 1$) | Ellipses horizontal, hyperbolas vertical | | **Hyperboloid 2 sheets** | $x^2 + y^2 - z^2 = -1$ (RHS $= -1$) | Two separate bowls, gap near origin | | **Elliptic Cone** | $z^2 = x^2 + y^2$ (RHS $= 0$) | Ellipses, two nappes meeting at vertex | | **Elliptic Cylinder** | $x^2 + y^2 = 1$ ($z$ missing) | Same ellipse forever | | **Parabolic Cylinder** | $x^2 = 4py$ ($z$ missing) | Same parabola forever | | **Hyperbolic Cylinder** | $x^2 - y^2 = 1$ ($z$ missing) | Same hyperbola forever |