Using String to Draw Concentric Circles
Using merely a circle and directly lines, it'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 plant in the world effectually the states. From an archery target to an apple, can you name these geometric shapes?
Y'all will by the terminate of this step-by-step tutorial, and you lot'll also be set up to try your hand at some geometric fine art yourself, which you could then spin into op art or cord fine art. To create the curve examples that follow, I used a pencil, ruler, and protractor to marking off degrees and keep things exact. If yous want, you lot could even use the free computer plan GeoGebra to draw concentric circles, ellipses, and cardioids.
Materials & Tools Needed
For all of these curves, nosotros'll demand a few basic geometry tools and cartoon utensils.
- paper
- ruler or straight border
- pen or pencil
- compass for cartoon circles (or images of circles or regular polygons)
- protractor (for marking off precise degrees)
Option 1. Making Concentric Circles
Concentric circles are circles that share a midpoint, such as an archery target or a dartboard. The circles, though different size, all have the same bullseye. Regular polygons, regular polyhedra, and spheres can also be described every bit concentric as they all share the same center. In fact, in our first example below, nosotros create a concentric circle by drawing polygons:
Step i: Mark a Circle at Fifty-fifty Intervals
This circle is marked every 10 degrees, then 36 marks total:
Stride 2: Connect Ane Marking to Another
The number of marks skipped will determine the size of the concentric circumvolve created. I chose to skip eight:
Footstep 3: Create Lines All the Way Around the Circle
Take the next mark and connect it to the marker alee of the one you connected the previous i to. Keep doing this.
Based on your first marker, accept the next mark and connect it to the marking ahead of the 1 y'all connected the previous one to, so go on doing this:
The resultant polygon has the same number of sides every bit there were marks on the original circle:
You can then use the within circle equally a starting point for creating some other circle:
These concentric circle designs are really creating star polygons, as discussed in the posts on creating star designs on pumpkins and creating torus knots. In the image below, using GeoGebra (Classic), I took a circle with 30 marks and connected them in a design with vi pentagrams. This works considering 30 / 5 = 6:
Option 2. Making Ellipses
Like the Earth'southward orbit around the Sun, an ellipse is a "closed curved shape that is flat" and best described as an oval. This squashed circle has two focal points, where "the sum of the distances to the foci is constant for every point on the curve."
Pace one: Mark Off a Circle by Degrees
There are lots of ways to create ellipses, merely this one is pretty fun. Start with a circle with a number of evenly spaced marks; I again chose ten degrees. I then connected 2 of the marks that were 180 degrees apart, halving the circumvolve. This line will become the major axis:
Stride 2: Choose a Indicate on the Major Axis
Marking a indicate on your major centrality. Making this point further from the center of the circle makes the ellipse longer and narrower.
Step three: Draw a Line from the Focal Betoken
Make a right bending on one of the marks on the circumvolve so that one of the sides of the bending goes through the focus. Connect the line from the mark to where it intersects the circle. Go along this procedure for all of the marks on the circle:
Halfway done; the resolution increases as it gets further from the focus:
And hither is the completed ellipse. Note you could ameliorate the resolution near the focus by repeating the process 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 post-obit middle shape, a design that brought me to mind the curve in mathematics that is known for its heart-like shape, the cardioid, which will cover side by side.
Pick iii. Making Cardioids
A cardioid is formed by a rolling circle's path across the circumference of some other circle, all the while keeping its radius the aforementioned. The term, offset used in 1741 by Giovanni de Castillon (in Philosophical Transactions of the Royal Society, 1741), originates from the Greek give-and-take for "heart." An everyday object that holds a true cardioid shape? An apple.
Step i: Mark a Circle'due south Circumference Evenly
To create a cardioid, offset with a circumvolve with a number of evenly spaced marks; I chose every 10 degrees on this i all the same again:
Stride 2: Skip Ahead by Two Marks
Starting with any one mark, connect a line from it to a mark that is ii marks away:
Stride iii: Count by Ones & Twos
Take the next mark after your starting point and connect information technology to the marking that is two across the ending point of the last marker.
Take the adjacent mark after your starting point and connect it to the mark that is 2 beyond the ending point of the final mark. Repeat. You are basically counting by ones on the starting points, and by twos on the ending points. Keep going and you'll brainstorm to see the bend take shape:
Stride four: Keep on Other Side
To avert confusion, simply focus on the marking you're working on: counting by ones for the starting betoken and past twos for the catastrophe signal:
Footstep 5: End at Original Starting Point
Y'all'll meet the heart take shape in one case you become completely around; stop once you render to the first marking. My completed cardioid:
You can follow this aforementioned process merely by counting by threes on the catastrophe marks or fours and you will get more abrupt points. If yous skip less often, you will become a more gradual spiral. The curve beneath was generated by counting by twos only every 4th fourth dimension.
Option 4. Combine Curves & Create a Heart
Inspired, I decided to endeavour and design a bend that looks more like a heart by combining linear sections, concentric circumvolve sections, and cardioid sections. I call up the final design looks pretty good. What practice you call up?
Step ane: Marking a Circle into Fours
To outset, you lot again mark a circle evenly. I chose once again 36 marks. I also fabricated the marks at ninety caste intervals more than visible; these will be important markers:
Step 2: Form the Heart's Point
From the bottom, describe a line between two of the marks at xc-degree intervals. This will become part of the pointy end of the eye:
Step three: Draw Lines for a Concentric Circumvolve
Connect the side by side marks as if yous were making the concentric circle design. Terminate when you get to the line that connects to the signal 180 degrees from the starting signal. You could do the aforementioned on the other side. You might not want to practice this until you are completely done.
Pace 4: Repeat Procedure on Other Side
Though nosotros go ahead and complete the concentric-circumvolve design for the reverse side, yous tin can cull to expect until you are completely done with one half in order to go on all the lines/marks direct:
Step 5: Add a Cardioid Department
Adjacent, you will add together a cardioid department, starting at the xc-degree marker correct of the heart's "betoken." From here, connect marks past counting past ones for the starting point of the lines, and by counting by twos for the catastrophe points:
Pace six: Repeat on Other Side
With the cardioid section applied to only half of the circle, you lot can already come across the middle forming. To end the "kardioedides," simply repeat the steps for the left side, and your heart is consummate!
Cool, right? Finally, from this exercise, I took on one more challenge: Using a computer 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 information technology comes to curves. Now that you lot're familiar with the basic mathematical curves of the concentric circle, ellipse, and cardioid, you, too, can try your ain design ideas. Every bit we've found, with a simple circle and straight lines (and some math logic), you tin can create any number of geometric shapes and combinations. What does your eye come across?
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Source: https://mathcraft.wonderhowto.com/how-to/create-concentric-circles-ellipses-cardioids-more-using-straight-lines-circles-0131356/