How to Draw Eccentric Circles

Using only a circle and straight lines, information technology's possible to create various artful curves that combine both art and mathematics. The geometry behind the concentric circle, ellipse, and cardioid dates back centuries and is easily found in the world effectually us. From an archery target to an apple, can you name these geometric shapes?

You volition by the end of this footstep-by-step tutorial, and you lot'll also be set up to try your mitt at some geometric fine art yourself, which yous could so spin into op fine art or string art. To create the curve examples that follow, I used a pencil, ruler, and protractor to mark off degrees and continue things exact. If you desire, y'all could even use the costless computer program GeoGebra to draw concentric circles, ellipses, and cardioids.

How to Create Concentric Circles, Ellipses, Cardioids & More Using Straight Lines & Circles — Concentric circles (superlative), ellipse (bottom left), and cardioid (lesser right). *Images past Cory Poole/WonderHowTo*

Materials & Tools Needed

For all of these curves, we'll demand a few basic geometry tools and cartoon utensils.

paper
ruler or direct edge
pen or pencil
compass for drawing circles (or images of circles or regular polygons)
protractor (for marking off precise degrees)

Pick i. Making Concentric Circles

Concentric circles are circles that share a midpoint, such every bit an archery target or a dartboard. The circles, though dissimilar size, all have the aforementioned bullseye. Regular polygons, regular polyhedra, and spheres tin likewise be described every bit concentric equally they all share the aforementioned heart. In fact, in our commencement example beneath, we create a concentric circle by drawing polygons:

Step 1: Mark a Circle at Even Intervals

This circle is marked every x degrees, so 36 marks full:

Pace 2: Connect One Mark to Another

The number of marks skipped will make up one's mind the size of the concentric circle created. I chose to skip eight:

Step 3: Create Lines All the Fashion Around the Circle

Take the next mark and connect it to the mark ahead of the one you connected the previous one to. Continue doing this.

Based on your get-go marker, take the next marking and connect it to the mark alee of the one y'all connected the previous one to, and so proceed doing this:

The resultant polygon has the same number of sides equally at that place were marks on the original circle:

Y'all can then use the within circle every bit a starting point for creating another circle:

These concentric circle designs are really creating star polygons, equally discussed in the posts on creating star designs on pumpkins and creating torus knots. In the paradigm below, using GeoGebra (Classic), I took a circle with xxx marks and connected them in a design with six pentagrams. This works considering thirty / 5 = vi:

Option 2. Making Ellipses

Similar the World'due south orbit around the Dominicus, an ellipse is a "airtight curved shape that is flat" and all-time described as an oval. This squashed circle has ii focal points, where "the sum of the distances to the foci is constant for every indicate on the bend."

Stride one: Mark Off a Circle by Degrees

In that location are lots of means to create ellipses, only this ane is pretty fun. Start with a circle with a number of evenly spaced marks; I again chose 10 degrees. I and so connected ii of the marks that were 180 degrees apart, halving the circle. This line will become the major axis:

Pace 2: Choose a Point on the Major Axis

Marker a point on your major axis. Making this point further from the eye of the circumvolve makes the ellipse longer and narrower.

Step 3: Draw a Line from the Focal Point

Brand a right bending on one of the marks on the circle then that 1 of the sides of the bending goes through the focus. Connect the line from the mark to where it intersects the circumvolve. Go on this process for all of the marks on the circle:

Halfway washed; the resolution increases equally it gets further from the focus:

And here is the completed ellipse. Notation you could improve the resolution near the focus by repeating the procedure using the focus on the other side:

From the ellipse, I went a step further and intersected two ellipses at correct angles. The result was the following centre shape, a design that brought me to mind the curve in mathematics that is known for its center-like shape, the cardioid, which will cover next.

Option three. Making Cardioids

A cardioid is formed by a rolling circle'south path across the circumference of some other circle, all the while keeping its radius the same. The term, kickoff used in 1741 by Giovanni de Castillon (in Philosophical Transactions of the Royal Club, 1741), originates from the Greek word for "heart." An everyday object that holds a true cardioid shape? An apple tree.

Step 1: Mark a Circumvolve's Circumference Evenly

To create a cardioid, start with a circle with a number of evenly spaced marks; I chose every 10 degrees on this one yet again:

Footstep two: Skip Alee by Ii Marks

Starting with any i marking, connect a line from it to a mark that is ii marks away:

Step 3: Count by Ones & Twos

Take the next marker after your starting point and connect it to the marking that is two beyond the catastrophe point of the final marker.

Have the next mark after your starting bespeak and connect it to the mark that is two beyond the ending point of the concluding marking. Repeat. You are basically counting by ones on the starting points, and by twos on the ending points. Proceed going and yous'll brainstorm to see the bend take shape:

Pace 4: Go on on Other Side

To avoid confusion, simply focus on the mark yous're working on: counting by ones for the starting bespeak and past twos for the ending bespeak:

Step five: Finish at Original Starting Point

You'll see the heart have shape once yous go completely around; stop once you render to the showtime mark. My completed cardioid:

You can follow this same process only by counting by threes on the catastrophe marks or fours and you will get more precipitous points. If you skip less often, y'all will get a more gradual spiral. The bend below was generated by counting past twos only every quaternary time.

Option iv. Combine Curves & Create a Heart

Inspired, I decided to effort and pattern a curve that looks more like a middle past combining linear sections, concentric circle sections, and cardioid sections. I think the final design looks pretty proficient. What do yous think?

Stride 1: Mark a Circle into Fours

To start, you once more mark a circumvolve evenly. I chose once again 36 marks. I as well made the marks at 90 degree intervals more visible; these will be important markers:

Step 2: Form the Heart's Point

From the bottom, depict a line between two of the marks at ninety-degree intervals. This will become part of the pointy terminate of the middle:

Step 3: Draw Lines for a Concentric Circle

Connect the next marks as if you were making the concentric circumvolve design. Stop when yous get to the line that connects to the point 180 degrees from the starting point. You lot could do the same on the other side. You might non desire to do this until you are completely done.

Step iv: Repeat Process on Other Side

Though we go alee and consummate the concentric-circle design for the opposite side, you can choose to await until you are completely done with 1 one-half in order to go along all the lines/marks straight:

Step 5: Add a Cardioid Section

Adjacent, yous volition add a cardioid department, starting at the ninety-caste mark correct of the center's "point." From hither, connect marks by counting by ones for the starting bespeak of the lines, and by counting by twos for the ending points:

Stride vi: Echo on Other Side

With the cardioid department applied to only half of the circle, you can already run across the heart forming. To finish the "kardioedides," simply repeat the steps for the left side, and your center is complete!

Cool, right? Finally, from this exercise, I took on ane more challenge: Using a figurer program, I attempted to create the same heart design using 59 marks instead of 36. I then colored it in:

The ideas are pretty fluid when it comes to curves. Now that you're familiar with the basic mathematical curves of the concentric circle, ellipse, and cardioid, yous, too, tin can try your own pattern ideas. Equally we've establish, with a simple circumvolve and directly lines (and some math logic), y'all tin create any number of geometric shapes and combinations. What does your heart come across?

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