Desmos Art: The Definitive Guide to Computational Sketching | Math Vault (2024)

If you’re of the type who enjoys playing around with graphing calculator, then you might be interested in the so-called Desmos Art. These are basicallypictures and animationscreated in Desmosprimarily through the clever use of equations and inequalities — among other features such as tables, animating sliders and regressionmodels.Indeed, if you go to this official staff pick page, youshouldsee that a whole bunch of creative artworks were already being created — anything fromcartoon characters,landscapeto logos and portraits.

But here’saproblem: when you reach a Desmos artwork page, youget to see the end-result with all the equations and inequalities, without necessarily having any clue about how the sketching process comes about from the beginning to the end:

• Why does this weird-looking equation pop out from nowhere?
• How did the author know that $3.982$ is the leading coefficient to use?
• And how did those numbers get so precise up to fourdecimals digits?

To remedy this situation, we came up with a laborious andinteresting solution: we decided to go out there and create a new Desmos sketch from scratch, and work backward to comment a bit onthe steps, procedures and reasoning involvedin the sketching process —as we go through the differentstages and portionsof the drawing.

And with that, let’s proceed straight into our Desmos Art guide in computational sketching, which — as you might have guessed — is going to be both fun and informative — even if you have no intention whatsoever of using any graphing calculator in the near future.

Step 1: Initial Setup — SourcePicture

Unless you are some kind of sketching guru — which we certainly are not — it’s generally better tobase thesketching on some underlying model: some picture that you actually want to draw; some picture thatyou can rely upon in case thesketching derailsby a hugemargin.

And in case it’snot clear enough, you should only choose a picture that you really, really enjoy sketching— something that you can derive awhole lot ofsatisfaction from. This is because unlike ordinary pencil-and-eraser sketching, sketching in Desmos has to be done byfilling the command lines with equations, tables and inequalities, so thatif you can’t fuel enough passion into a drawing, then the time and effort invested into reproducing itisprobably not going to worth it.

Also, when choosing a picture, take good care to select one that is not too difficult to draw. Otherwise, you might have to regret your decision much later when you’re already, say, 21 hours into the process. Of course, youdon’t want to make the picture too easy either, as it might not be much of a gratifying experience if youdoitthatway.

In our case, using theReddit Goldmembership recently awarded to us, we went out there and createdour first very ownsignature Reddit alien avatar:

For the lack of better terms, let’s just call these lovely creatures Math Vault Redditlady and Redditdog, respectively. Now,does this logo make a good choice as a source image? We certainly think it does:

• Passion Level: Since we only create avatars that we like, channeling energy into recreating the creatures invented in part by ourselves is definitely not an issue.
• Complexity: Both the Redditlady and the Redditdog have verywell-defined borders and colors,making it a formidablechallenge that’s nottoo intimidating.

And with both criteria passed, we proceed to happily importour avatar image into Desmos. By default, Desmos likes to set the origin $(0,0)$ as the center of the image, and that is a good choice for most purposes. However, depending on the size of thesource image, sometimes it makes sense to consider rescaling it so that at the default zoom level, the image occupies around 70% of the graphing grideither in height or in width. If the image is too small, then it’s hard to appreciate anyfine detail, and if it’s too big, then onemighthave a hard time focusing on the big picture — literally.

In our case, since theavatar isjust about the right size to start with, we decided to keep the dimension of the image to the default $10 \times 10$ setting at the end. We also didn’t alterthe image’saspect ratioeither, which in many cases aren’t reallynecessary anyway.

But just to drive the message home a bit, setting up asource image correctly from the get-go is more important than it seems, since the last thing wewant to be doing is to, say,rescale the image to an appropriate size after beingalready halfway through the sketching process.

Step 2: Divide and Conquer

How do you deal with a huge task that seems impossible to complete? You partition the task into smaller subtasks. That is, tasks that are so small that you actually enjoy doing them!

In particular, when it comes to sketching in Desmos, you wantto section the picture into different portions, each of which can befinished in a reasonable amount of time, sothat when it comes to the time ofsketching, you focus on onesingle portion and onesingleportion only.

In our case, since we know that sketching the entire Redditlady and its companion pet can be quite a challenge, instead of resorting to despair, we divide the entire avatar into 6 portions — from the top to the bottom:

By inspection, while we see that most of these portions arerelatively simpleto implement, we still anticipate some challenges ahead of us when it comes to theArmor portion. In an attempt to make the Armor less intimidating, we proceed tosubdivide itfurther into 7subportions:

With the gist of ourorganisational structure now settled, we then proceed to create aDesmos folderfor each of the general portions, all of the while making a mental note that every command line pertaining to a specificportion shouldbe dragged into itsassociated folder — if it’s not already in there.

And while Desmos doesn’t really support folder nesting at this point, there’s nothing preventing us from usingcomment lines to provide labels for the subportions. True, while this level of organization couldbe an overkill for a simple picture, as the complexityof thepicture increases, structuring the command lines this way can makeline identification and future reference substantially easier.

In our case for example, the need for quick reference became so urgent that towardsthe end of the sketching, we ended up creating far more subportions than we originally thought needed. Here’s our updated organisational structure at the completion of theavatar:

(Yep. The Redditdog proved to be a bit more annoying than originally anticipated. Hence the saying “Don’t judge a puppy by its cover.”)

Learn to sketch a puppy computationally in Desmos through equations and inequalities!Click to Tweet

Step 3: Portion Crunching

As mentioned a bit earlier, when working on a sketch, it’s often less intimidating tojust focus on a portion while ignoring the rest. We call this portion crunching, andis done in part to ensure that the work completedwith minimal resistanceand mental blocks. In Desmos, portion crunchingusually boils down tothefollowing two parts: curving drawing and region coloring.

Step 3a: Drawing Curves

In general, a self-contained portion of an image is composed of close figures, which in turn can be construed asa series of curve segments. However, since these curvesegmentstendinteract with each other, it would be hard for us to draw these segments sequentially— as would be done in ordinary pencil-and-eraser sketching.Instead, we prefer to resort to the strategyas outlined below:

1. Model a curve segment using an appropriate class of equations, and leave itsgraphwith thefull domain intact.
2. Do the modeling for eachof the curve segments in the close figure.
3. Cut off the excessive parts ofeach curve — by imposing restrictions on theassociated equation.

In particular, wewant the curve trimming tohappen at the very end, becauseonce the curve modeling are done for each curve segment ina close figure, Desmos will map out allthe intersection pointsalong withtheir coordinates — the latter of which provide key information as to whereto cut off thecurves, and might not be available hadwedecided to trim the curves off from the get-go.

In actuality though, we’llprobably spend more time modeling the curve segments rather than trimmingthem— as there aresimply too manycurvesin too manydifferent shapes and forms. And while it might be hardto provide an exhaustive set of tips for modeling each curve segment we will ever encounter, the following proceduresshould be more than enough totake care ofmost of them.

Modeling Line Segments

Many close figures are composed of nothing but lines, and vertical/horizontal line segments are about as easy as onecan get. For example, to create the line segment connecting $(1.2, 5)$ with$(1.2, 5.67)$, simply typing $x=1.2 \, \{5\le y \le 5.67 \}$ into a command line will do just fine.

In general, if you know the coordinates of the two endpoints of a line segment, then you can enter them into a Desmos table, and re-configure the points (via the Gear Icon) to have a line segment passing through it.

On the other hand, if you only have the coordinate of a single point on the line segment, say $(2.33,4.67)$, then you can still resort tothe point-slope form, which in this case becomes $y=(x-2.33)+4.67$. In fact, by simply adjusting the slopeone digit at a time, and observing the graph at an ever-increasing zoom level, it shouldn’t take long toobtain a line thatreasonablyoverlaps with the segment you are trying to model.

Modeling Circular/Elliptical Segments

In some cases, the borders of objects(e.g., eyes, ring, hands) are either ellipsesthemselves, or can modeled usingsome elliptical arcs. In which case, the following modeling procedure could be useful:

Modeling Other Curvy Segments

While elliptical segments tend to be curves with a rather pronounced curvature, there are other curve segments such that — while still curvy in their own way — does not posses this distinct characteristic. In which case, what we’ve found is that by modeling them using “polynomials”, it’s possible to sneak in some reasonable fit most of the time.

For example, to model a concave-up, increasing curve segment with a vertex close to, say, $(1.56,6,78)$, we can start withthe vertex form $y = (x-1.56)^2 + 6.78$. And if the segment has a slower increase in the beginning but a faster increase in the end, the vertex form of a quartic function — say $y=(x-1.56)^4 + 6.78$ — can be used as a starting point instead.

In fact, by adjusting the “degree” of the “polynomial”to any other positive realnumber, we can provide a reasonable fit to a surprising number ofcurvy segments as well:

• For a U-curve segment, degree $4$ or more usually works wonder.
• For a concave up, line-like segment, a degree between $1$ and $2$ seems to be more appropriate.
• For a concave-down, increasing curve segment, a degree between $0$ and $1$ can be used instead.

In fact, by applying horizontal or vertical reflections (e.g., $-(x-1.56)^4 + 6.78$, $[-(x-1.56)]^{1.5} + 6.78$) to a “polynomial” vertex form, we can usually bend the resulting graph to any direction we want. This flexibility would result in the following procedure for modeling non-elliptical curve segments in general:

Subdividing Curve Segments

In some occasions, a curve segment canbe a bit convoluted that it’s simply easier to subdivide it into multiple curve segments instead (as opposed tousinghigher-order polynomial regression). For example:

• A round corner can be conceived as three curve segments, consisting of the two surrounding lines and aquarter-circle for the edge.
• Thetop portion of a mustache can sometimes be broken down into two elliptical segments.
• The tail of a pet can sometimes bemadeinto a seriesof line/curve segments, and perhaps with a half-circlereserved for the tip of the tail.

In general, a curve segment can be cut by visualizing vertical lines in between. In which case, we can model the resulting subsegments individually as functions of $x$ — and if needed to — groupthese functionstogether again by defining a piecewise functionin terms of $x$. Alternatively, a curve segment can also be cut by visualizinghorizontal lines in between. In which case, the resulting subsegments canbe modeled individually as functions of $y$ instead.

(For more on defining piecewise functions in Desmos, see Desmos: A Definitive Guide on Graphing and Computing.)

Curve Trimming

Once we finish modeling all the curve segments in a close figure, we can begin totrim off the excessive parts of eachmodeling curve by imposing restrictions on the associated equation. For most purposes, thisis usually done by specifying the lower/upper bounds within which the curve is supposed to occur. For example:

• If a half-circle is only to be graphed when $x$ is between $2.57$ and $13.534$, then adding the clause $\{ 2.57 \le x \le 13.534 \}$ to the end of itsequation will achieve the desired effect.
• If a U-curve is only to be displayed when $y$ is between $-2.3$ and $1.87$, then adding $\{ -2.3 \le y \le 1.87 \}$ to the end of its equation willtake care of it.

As alluded to a bit earlier, theactual numbers fortheupper/lower bounds are usually part of the coordinates of someintersection point. whichcan be displayed up to four decimals if you zoom in hard enough before clicking on them. In the case where anupper/lower bound doesn’t occur at any of the key points, the bound will have to be obtained manually byclicking a bit on the region where the curve is supposed to be trimmedoff.

In some occasions (such as the case withan obliquely-trimmed ellipse), a simple upper/lower bound simply won’t be enough to cut the curve the right way. Instead, you will have to define the upper/lower bounds as actualfunctions (either in terms of $x$ or in terms of $y$), and use them accordingly inthe restriction clauses. In otheroccasions, you might even have to stack up multiple restriction clauses in order to get the desired effect.

Step 3b: Coloring Regions

After modeling and trimming all the curve segments ina close figure, it’s about time to start thinking about coloring the region inside. For implicit equations such asthat of a full circle or an ellipse, this can usually be done by replacing the $=$ sign in the equation with the $\le$ sign. In most likelihood though, coloring a region is not going this easy, as manyclose figures have anirregular areathat is bounded by a series of different-looking curve segments instead.

For example. in the case where an irregular area occurs but has ratherwell-defined upper and lower bounds, the coloring is usually done using the procedure below:

1. Define the upper bound of the region as a function of $x$ (let’s call it$g(x)$). If the upper bound consists of multiple curve segments, then the function will haveto be defined piecewisein order to take into account all the segments involved.
2. In a similar manner, define the lower bound of the region as a function of $x$ as well (let’s call it $f(x)$ for now).
3. Color the region by enteringthe inequality $f(x) \le y \le g(x)$ into a command line.

In some occasions, it might not be possible to color a region using $y$ as the bounded variable. If that’s the case, then you might have to adjust the aboveprocedure to use $x$ as the bounded variable instead (e.g., color the region via an inequality of the form $f(y) \le x \le g(y)$).

In Desmos, if youcolor a region using a single inequality, the coloring will usually be a bit on the paler side. To increase the depth of the coloring, one thing you can do is to simply duplicate the same inequality three or more timesin the command line. In fact, if you have some knowledge in color mixing, then you shouldplay around with the different color combinations a bitand see what kind of end result you get!

So, how exactly does portion crunching work out in practice? To figure that out, we thought wewouldelaboratea bit on howwe manage to crunch out the Antenna & Hat portionof our avatar.

First of all, we would start by subdividing the antennainto several sections, andmodeled them individually using a series of functions:

• Antenna Ring: This is simply modeled using two full circles.
• Antenna Cable (Upper Part): Being more like a round corner, the cable is segmented and modeled using a line, a half-circle and another line joining the ring to the hat.
• Antenna Cable (Lower Part): This is simply modeled using two lines.

As for the hat, here’s what we did:

• Hat Disks (Top and Bottom): Sorry. We cheated. Each of the disks ismodeled after anrotated ellipse, which takes on a slightly more involved equation.
• Hat Borders (Vertical): These are simply modeled aftertwo oblique lines.
• HatMid-lines(Horizontal Curves): These are modeled using quadratic functions in vertex form.

And with the modeling alldone, we’re left with a bunch of curves that intersect each other. Using the coordinates of these intersection points, we proceed to trim off the excessive parts of the curves one by one:

• Outer Circle (Antenna Ring): This circle needs to havea gap on the upper-left end, so we trimmed off the excess by:
  1. Defining a linear function connecting the endpoints of the gap (let’s callit $f(x)$ for now).
  2. Appending the restriction clause $\{ y \le f(x) \}$ tothe end of theequation ofthe outer circle.
• Antenna Cable: Each equation making up the antenna cable is trimmed off by imposing simple restriction clauses on $x$ (e.g., $\{0.6153 \le x \le 1.755 \}$), where the actual numbers for the upper/lower bounds are obtained through the coordinates of the intersection points.
• Hat: the equations for the vertical hat borders and the horizontal hat mid-lines are trimmed off in a similar manner — again using the relevant intersection points.

Once the curves are well-trimmed, we then proceed to color the different parts of the hat as follows:

• Top Disk: This is colored easily by changing the $=$ sign in the equation to $\le$.
• Upper and Lower Strip: Each of the strip is colored using an inequality of the form $f(x) \le y \le g(x)$, where $f(x)$ and $g(x)$ are the lower and upper bound functions of the strip region, respectively, and both functions have to be defined piecewise(see the figure below for more).
• Bottom Disk: Since the Lower Strip is red and this part needs to be in orange, we color it by appending the restriction clause $\{ y\le f(x) \}$ into the equation of the ellipse.

All right. So that pretty much concludes thevery first portion of our avatar! By the way, here’s a visual depiction of the upper/lower bounds for the red Lower Strip region — in case itstill hasn’t clicked yet:

Step 4: MorePortion Crunching

Once you run through all the hurdles of drawing a portion/subportion of yourpicture, it’s time to move on to other portions and put your new-foundskills into good use. How? By picking a close figure, focusing on modeling the curves, trimming off the excess and coloring the figure, and before you know it,everything is set in front of your eyes!

(after 6 hours of undivided attention, for example)

However, there’s a caveat: for many aspiring Desmos artists,this advice doesn’t seem to be particularly actionable.If that’s you, then you’ll definitively find thefollowing tipsand strategies we used to sketchthe other portions of theavatar a bit more informative.

Redditlady’s Face

As expected, we find the faceto be one of the easiest portions to sketch, since most ofthe curve segments involved are either full circlesthemselves, or parts of a circle/ellipse:

• Eyes: As round as they are, these eyes can be easily modeled using full circles. Indeed, with just a bit of parameter tweaking, it didn’t take long before we figure out the radius and the center of the circle, which is then coloredby changing the $=$ sign intheirequation to $\le$. The resulting inequalities are then duplicated several times to increase the depth of the coloring.
• Face Border & Smile: Beingboth elliptical segments in disguise, the face border and the smile are both modeled usingfull ellipses. The “smile” ellipse is trimmed off usingtherestriction clause $\{ y \le 0.202 \}$, while the face border had to be trimmed off using a custom-defined upper bound function on $y$ (the latter of which isdefined as the line connecting the two points where the face and the hat intersect).
• Left Ear: The two outer circular arcs are modeled using full circles, while the innermost curve — which resembles a line — is actually modeled using the vertex form of a “polynomial” of degree $0.7$ (i.e., $c(x-h)^{0.7}+k$). Using the coordinates of the relevant intersection points provided by Desmos, the three resulting curves are then trimmed off by appending simple upper/lower bounds on their respective equations.
• Right Ear: The two outer arcs are now modeled aftera full circle (inner arc) and a full ellipse (outermost arc), respectively. The two inner curves are not exactly elliptical in nature, so we modeled the top one using a quadratic vertex form($0.24 (x-2.783)^2+2.188$ to be sure), and the bottom one using a “polynomial” vertex form with degree $2.7$ (after a bit of tweaking). Once there, by using the coordinates of the surrounding intersection points, we trimmed off the two outer arcs by imposing simple upper/lower bounds on both $x$ and $y$, andthe two inner curves by appending simple restriction clauses on the value of $x$.

Being composed of simple geometrical shapes, we didn’t anticipate the Hand & Rubik’s Cube portion to be particular hard either, and in retrospect, we were almost entirely right on that one:

Hand: These are basically just two standard full ellipses— withno trimming required.

Rubik’s Cube (Left and Right Side): Since this partis composed entirely of lines, we basically wentpoint-slope formall the way through. In addition, since the vertical-looking lines here are indeedvertical, it didn’t take a lot of tweaking to finish this part.

Rubik’s Cube (Top Side): Same drill. Except that all the lines are now oblique, hence literallypoint-slope form all the way through.

So all seems to go well, until it came the time to color the cube that is! As youcan see in the left figure, the red portion is colored by defining $g_1(x)$ and $g_2(x)$ as the upper and lower bound functions, and by using the inequality $g_2(x) \le y$ $\le g_1(x)$ — which is duplicated five times in the command line to provide that reddish tint.

(by the way, these are not the actual names we gave to those functions, but ithelps tothink of them that way.)

In a similar manner, the left side of the cube iscolored by restricting the $y$-values to be between $f_2(x)$ and $f_1(x)$ — both of which had to bedefined piecewise. For example, the lower bound function $f_2(x)$ had to be defined by conjoining the left vertical line segment, the top-half of the ellipse, and the right vertical line segment— in that order.This of course means that the original equation of the full ellipse had to be reduced to a function first — before we canused it to define a piecewise function.

So in overall,while the coloring process did get a bit involved in the end, sketching the Rubik’s cube in Desmos is kind of cool in itself, asit exemplifies how a 3D figurecan be reproducedon a 2D interface — withequations and inequalities even!

While this portion might look simple at the first sight, it is actually a bit stickyas there are many different kinds of curve segments involved. But nevertheless, let’s have the picture do the talk for us first:

As can be seen above, the skirt alone is surrounded by seven curve segments, where:

• Curve 1 is modeled using the bottom-half of an ellipse.
• Curve2, 3 and 4 are modeled after quadratic vertex forms.
• Curve 6 is just a horizontal line — nothing fancy here.
• Curve5 and 7 are bothparts of the bottom half of the same ellipse.

With these seven curve segmentsmodeled, we then use the coordinates of the resulting intersection points to trim the curvesinto the right length. After which, we proceed todefine the upper bound function $g(x)$ by conjoining Curve 1,2,3 and 4, and the lower bound function $f(x)$ by conjoining Curve 5, 6 and 7. The skirt is then colored green using the inequality $f(x) \le y \le g(x)$, which is duplicated three times in the command line to make the skirt a bit greener.

As for the legs, we’ve decided to sketch them using the following modeling scheme instead:

• Curve c and Curve 0 are both modeled after the top-half of an ellipse.
• Curve a is modeled using a quadratic vertex form, while Curve b — which looks more like a “curvy line” — is modeled using a “polynomial” vertex form with degree $0.8$.
• Curve 8 and 9 got a bit more subtle, so we chose to model them using “polynomial” vertex form as well. After a bit of tweaking, we found that Curve 8 — which looks like an intermediate function between a line and a parabola — can be modeled fairly well withdegree $1.35$. As for Curve 9, we’ve found a match when the degree is $1.99$

(Yep. Looks like we’ve “curved” it a little bit.)

Redditdog

Remember us complaining about theRedditdog? Well, it’s about time to figure out just how annoying itwas!

The subportion we’re looking at here are the tail and the ears. As surprising as it seems, we didn’t model the tail using a bunch of lines and half-circles (which we probably should have). Instead, we decided to model the left and right sides of the tail using hyperbolas, and the tip of the tail using the top-half of a circle.

And yes, we had to dosome serious zooming and number tweaking to make sure that the parameters wererobust enough, and that we can actuallyglue the tail together without any human noticing. But that stilldoesn’t change the fact that the tail is really three segments in one. Not good if you are this dog of course! 🙂

As for the ears, since all the four outer borders are prettycurvy, we decided to modeled them after the top-half/bottom-half of an ellipse. On the other hand, becausethe inner borders are much less curvy thanthe outer ones, we decided finally to use “polynomial” vertex forms on them instead.

Face

All right. Let’s take a look at how we dealt with this, hmm… dogface:

• Face Border: This is just a plain-old full ellipse— with no trimming required.
• Eyes: Unlike the Redditlady, our Redditdog’s eyes are not exactly madeof full circles. As such, we have to take into account of the line segments in the upper-extreme cornerswhen modeling the borders. In order tocolor the eyes, we define the upper bound function $g(x)$piecewise by conjoining the line segment with the corresponding quarter-circle.We also definethe lower bound function $f(x)$ asthe bottom-half of the relevant circle. After which, each eye is then shaded red by duplicating the inequality $f(x) \le y \le g(x)$ three times.
• Nose: This is nothing more than a triangle, or more specifically,a horizontal line segment on the top followed by two oblique line segments at the bottom. Make sure you know by now how to sketch thesesegmentsand color the region within!
• Whiskers: Due to their curvy nature, we decided to model the whiskers afterthe bottom-half of a circle. In fact, since the whiskers are symmetric about a horizontal line, all we had to do is to work out the function for the right whisker, and apply a horizontal reflection to get the left one for free.

Apparently, someone on Reddit made sure that a lot of design go into a dog’s lower body. However, after a bit of trial-and-errors, we found that we were still able to model each of the curve segments usingeithera full ellipse, a half-ellipse, or a “polynomial” vertex form. For example:

• The two outermost curve segments on the left and right sides of the body can be modeled as full ellipses, which are thentrimmed off by imposing simple restrictions on both $x$ and $y$.
• The remaining curve segments forming the left and right “triangles” — which arenot ascurvy than the ones above — are modeled after “polynomial” vertex formwith varying degrees.
• The left and right “horseshoes”can be easily modeled using a combination of full ellipses and horizontal lines, each of which subject to some simple restriction clause.
• Similarly, the “trapezoidal” gap between the legs canalso be modeled using a combination ofhalf-ellipsesand horizontal lines— which are againsubject to some simple restriction clauses.

Redditlady’s Armor

As one would have guessed, the armor piece is probably the toughest portion we have had todeal with. To ease the sketching process, we divide it further into four subportions: the shoulders, the arm, the chest, and the “abdomen”.

Shoulders

First, here’s a graphical depiction of the breakdown of theshoulders and the “collarbone”:

As you can see, the curve segments here start to get a bit into the crazy zone. The left shoulder, for example. is colored by using Curve 1, 2, 3 jointly as the upper boundand Curve 4 as the lower bound. Curve 1 — being the round edge of the shoulder — is modeled after the top-half of a circle, while Curve 2 and Curve 4 are modeled using “polynomial” vertex forms (with degree $1.6$ and $4$, respectively). On the other hand, Curve 3,which is a part of the face border,had to be remodeledas the bottom-half of an ellipse — by recycling the equation of thefull ellipse which was previously used to model the face border.

In a similar manner, we also colored the right shouldergrey by using Curve a, b, c jointly as the upper bound, and Curve 9, f, e, das the lower bound, except that this time, the curve segments are modeled using a combination of lines (Curve f and e), half-circles/ellipses (Curve b, c and d) and “polynomial” vertex forms (Curve 9 and b). In fact, a similar modeling and coloring scheme applies tothe “collarbone” area as well.

Arm

Looking at the figure on the left, we see that the arm is a relativelyeasy portion to implement. For example:

• The sleevecanbe colored grey using an inequality of the form $f(x) \le y \le g(x)$, where $f(x)$ is the horizontal line used to modeled Curve 4, and $g(x)$ is the piecewise function formed by conjoining Curve 1,2 and 3 — all of which can be modeled after “polynomial” vertex forms.
• The two outer curve segments for the arm (i.e., Curve 5 and Curve a) are actually ellipticalarcs, and thus are modeled after full ellipses.
• On the other hand, Curve c and Curve 6 — which are abit more linear in nature — had to be modeled using “polynomial” vertex formsof low degree instead.
• The other remaining curvesare just line segments,so it didn’t take us a long time to reproduce them.

Chest

OK. Here comes the inevitable part: the chest. To begin, here’s a graphical depiction of most of the curve segments involved:

By inspection, the easiest part is probably the “tie” area in the middle, since it’s composed of a “trapezoid” on the top, and a “rectangle” at the bottom. To be sure,we did realize that neitherCurve 1 nor Curve 7 areactually lines, so we took good care tomodelthemusing “polynomial” vertex forms instead.

After that,we moved on to the second easiest parts, which we determined to be the tiny “trapezoid” on the upper-right, and the weird-looking region next to the hand. Why the second easiest? Because both regions can be colored using an inequality of the form $f(x) \le y \le g(x)$ — as long as we take good care indefining the upper and the lower bound function $f$ and $g$ to take into account all the relevant segments involved.

Now, here comes the stickypart: if you look at the region in the above figure that is enclosed in orange and blue, then it shouldn’t take long to realize that it would be difficult to colorthis regionusing standard inequalities of the form $f(x) \le y \le g(x)$. In fact, if you decide to stubbornlystick withthis route anyway, then you willend up “scratching”ourRedditlady’s armor with a few unsightly little patches!

Instead, what wewant to do is to color the region using an inequality of the form $f(y) \le x \le g(y)$, where thelower bound function $f(y)$ is defined by conjoining the relevant part of Curve 7, Curve c and Curve b, and where the upper bound function $g(y)$isdefined by conjoining Curve dwith the relevant part of Curve 2. Basically, $f(x)$ corresponds tothe curve in the above figurethat is outlined in orange, and $g(x)$ the part inblue.

In fact, byusing the same strategy, the two remaining areas in the chest region — the right“lung” and the left “outer lung”that is — can also be colored using a single inequality as well.

“Abdomen”

All right. We’ve come to the last part, which isnot too bad either — especially after going through the nitty-gritty of that Chest portion.

In a nutshell, we basically colored the abdomen in grey by using a standard inequality of the form $f(x) \le y \le g(x)$, where:

• $g(x)$ is the function we used to model Curve 1 (a “polynomial” vertex form with degree $1.6$ to be exact — due to the linearnature of the curve).
• $f(x)$ is the lower bound function obtained by conjoining — in that order — Curve 2, the relevant section of Curve 3, Curve 4, the relevant section of Curve 5, and finally, Curve 6.Interestingly, each of these curve segments can be modeled using “polynomial” vertex forms — in particular as quadratic functions with varying slopes.

And guess what? We’re done! All the 219 equations/inequalities that is! Here’s a before-and-after picturefor to be sure:

Granted, there’s still a lotof subtleties and nuances going into how each of theequations and inequalities are constructed. and that is why we’ve decided to reserve the best for last. Namely, if you’re still curious about all thetechnical details behind the sketch, then you should definitely go toour Redditlady’s Desmos page and take a look at those command lines yourself!

Closing Words

Wow!That was a lot of work and fun for us, so hopefully the same is true for you too. If anything, this is definitely graphic design of a different caliber — andwith a wicked geeky twisteven!

Indeed, sketching inDesmos is like drawing with twohands (or maybe the legs as well) tied, even though a voicedeep inside us is sayingthat thisis only good for the cause of mathematics. This is because by the time you become a Desmos sketching master, you would have acquired a solid intuition about various geometrical shapes and the corresponding equations/inequalitiesyou can use to model them. Who knows. perhaps it mighteven improve your ability to evaluate a double/triple integral — if you ever decide to take the multivariate calculus route that is!

All right. Ready for the long-awaited interactive table? Because here it is!

And there you have it. The end of yet another intensely graphical saga with Desmos! Now, back to you: what kind of picture are you looking to draw with equations and inequalities? Either way, make sure to make use of what you’ve just learned on curve modelling so that when the time comes, you can draw like a real computational artist!