Introduction: Construct Curves to Smoothly Connect Lines at Any Angle

One of the steps when building my son's Optimist dinghy (similar to this 'ible) was to attach "knees" in all corners. Here the task was to find/construct a curve which neatly "connects" side and front edges. Now what can be done to acomplish this:

  1. grab all round plates, pots, coffee and corned beef cans you can find and look if one of their diameters is roughly what you need
  2. construct a curve which exactly does what you want

I went for (2) by the following steps ...

Step 1: Mark Edges

First step is to mark the edges which should be connected by a curve.

In my case I chose to do this "in situ", but the method will be the same if you do this on a sheet of paper. What needs to be found are

  1. the angle at which two lines intersect
  2. start and endpoints of the connecting curve on these two lines

Working "in situ" has the advantage that the two lines can be found easily by holding a ruler against the edges and just extending the edge line. The curve enpoints were chosen to lie on the very edges of the plates.

Step 2: Divide

Next is to divide the lines between curve endpoints and intersection into equal pieces. My lines were measured to (roughly ;-) 100 and 130mm so I chose to divide them in 10 equal spaces of 10 and 13mm, respectively.

As you can see on the 2nd picture, the 10th divider of the vertical line isn't "exactly" meeting the intersection with the other edge line due to the fact that the measure was 103mm - a difference I chose to neglect - the error is small enough in practice.

Step 3: Draw Tangents

Last step is to draw the curve tangents in the following way

  1. You start with a tangent from 1st division mark after endpoint on 1st edge line to the first mark from intersection on the 2nd edge line
  2. You advance from mark to mark until you have connected last mark before intersection on 1st edge line to last mark before endpoint on 2nd edge line

E voila - you have just created a set of tangents along which you can cut / file / sand to obtain a nice curve connecting two edges at any desired angle.

Step 4: For the Curious - Background

For those of you who are curious: mathematically speaking this curve is not part of a circle but a parabolic curve.

Let's compare this parabolic line with a circle

case 1: edges are of equal length

  • circle center can be found by intersecting perpendicular lines through the edge endpoints
  • circle removes slightly less material
  • circle radius equals edge length only for 90° angle

case 2: edges are of different length

  • circle can be found by intersecting perpendicular lines at distances equal to the length of the shorter edge
  • circle meets the shorter edge but not the longer edge

note: pictures in this section show

  • black: parabola construction
  • blue: construction of circle center
  • red: best fitting circle

I'm not commenting on the aestethic aspect here ... personal point of view

I hope you like this idea as much as I liked to construct my "knees" according to this method and document it. Kind regards Mike

Comments

author
TwelveFoot made it! (author)2017-06-02

This may be my favorite instructable. I use this all the time now.

IMG_1228.JPG
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MikeD50 made it! (author)MikeD502017-06-06

Awesome !!! You took it to the next level by concatenating left and right bends. Congratulations and thank you for the feedback.

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fwincek made it! (author)2016-06-10

I have made parabolas of any size by using 'ball' chain, the stuff of old lamp and fan switches, a couple nails, and a flat vertical surface. You can buy pull-chain by-the-foot at a good hardware store in different sizes. I've made beautiful doorway arches from 10' of ball-chain, each end nailed to a flat, vertical wall about 2' apart and allowed the rest to droop; there's your form. Trace that curve and you have all your math work done by gravity. Beautiful and perfect every time!!

author
JamesL280 made it! (author)JamesL2802016-07-15

Actually, what you are describing is a catenary curve, not quite a parabola, but close. It is considered to be the ideal shape for structural arches. "The word catenary is derived from the Latin word catena, which means "chain". The English word catenary is usually attributed to Thomas Jefferson..." You are right on to use this method for your arches.

author
fwincek made it! (author)fwincek2016-07-16

Thank you!! Now I can really sound smart when I mention this method!!
Fran

author
Ellystu made it! (author)2016-06-07

So useful. And fascinating. I like the extra lesson on circle v parabolic curve. Thanks!

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MikeD50 made it! (author)MikeD502016-06-09

I have added a 4th step which you may want to review for more on circle vs. parabola

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Ellystu made it! (author)Ellystu2016-06-09

Thank you!

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bpark1000 made it! (author)2016-06-09

I looked up the wiki link cited, and this curve is a Bezier curve. It is a cubic curve. Bezier curves find application in "smooth curves" for type fonts and art, and roller coaster tracks. When used for the latter, they minimize the "jerk" in the ride.

author
alzie made it! (author)2016-06-08

Compass?

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MikeD50 made it! (author)MikeD502016-06-09

I have added a bit more background to compare with compass as step 4

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falcon13 made it! (author)2016-06-08

Thanks for sharing--this will come in handy for LOTS of projects that I do.

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Zaphod Beetlebrox made it! (author)2016-06-06

Very cool thanks for sharing, have you entered any contests?

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MikeD50 made it! (author)MikeD502016-06-08

nope - I think it's not my aim to contest ... I am happy with sharing per se ;-)

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Zaphod Beetlebrox made it! (author)Zaphod Beetlebrox2016-06-08

I just wanted to know if I should check back in a few days once you where accepted into a contest :) Keep up the good work.

author
askjerry made it! (author)2016-06-08

Very nicely done sir!

Yup... I would have grabbed the coffee can... but now... no... now I've learned a new trick.

Thanks!

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TeresaM7 made it! (author)2016-06-07

For goodness sake, how is it I've never seen this before? So silly simple, too. Thank you, MikeD50!

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MikeD50 made it! (author)MikeD502016-06-08

maybe because you didn't look into boat building school books of 1920 *LOL*

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TeresaM7 made it! (author)TeresaM72016-06-08

True, but I grew up on a farm in the mid 60's though early 80's. I've been building stuff since I can remember. Never seen this, though! It's going to come in very handy, I can tell you that. :)

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Crumpred made it! (author)2016-06-07

How cool!

I think that this would also be useful for perspective drawing

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MikeD50 made it! (author)MikeD502016-06-08

hmm ... perspective drawing (of circles and circular arcs) require ellipses or elliptic arcs. Look under "conjugate diameter" and "Rytz"

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steynvdw made it! (author)2016-06-07

Can one still see the starting points of the cuts after the first cut?

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MikeD50 made it! (author)MikeD502016-06-08

as I used different parts of wood - yes ... the grain changes and I have a tiny little gap where the pieces touch each other ... as far as the curve itself is concerned - no ... as the slope of the curve gradually and slowly changes from the one edge towards the other.

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WesleyC1 made it! (author)2016-06-08

Parabolic FTW!

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WesleyC1 made it! (author)2016-06-08

Parabolic FTW!

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chuckstake made it! (author)2016-06-08

thank you

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jdaniels20 made it! (author)2016-06-08

WOW!

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TwelveFoot made it! (author)2016-06-07

Am I the only one whose previous method was to draw an arc with a compass, not find something round.

This is definitely my new method, the parabolic curve is so classy.

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bevin.chu made it! (author)2016-06-07

Good tip!

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throbscottle made it! (author)2016-06-07

Beats the pants off freehand!

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throbscottle made it! (author)2016-06-07

Beats the pants off freehand!

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николайм made it! (author)2016-06-07

кривые Безье строятся по похожему принципу

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brids78 made it! (author)brids782016-06-07

So True!

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Wild-Bill made it! (author)2016-06-07

Thank You - for all my years I have been making my curves conical or free hand. That is such a pretty curve. Oh, it is not a parabolic curve.

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marcsulf made it! (author)marcsulf2016-06-07

Yes, it is a parabolic curve. Please see this wiki link for an explanation:

https://en.wikipedia.org/wiki/Envelope_(mathematics)#Example_2

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Wild-Bill made it! (author)Wild-Bill2016-06-07

Ah, rotated through 45 degrees. I have a project I am in the midst of doing, that I am now planning to use the technique.

author
fjr_scoot made it! (author)2016-06-07

As a technically-oriented person, I very much appreciate your sharing
this - for those situations when tracing the curve of a piece of uniformly "bendy" material (like
a thin strip of formica) aligned with both sides - just won't do.

author
nanaverm made it! (author)2016-06-07

Reminds me of string art! But a great idea.

author
nanaverm made it! (author)2016-06-07

Reminds me of string art! But a great idea.

author
nanaverm made it! (author)2016-06-07

Reminds me of string art! But a great idea.

author
The Rambler made it! (author)2016-06-07

Haha, I do this all the time! It makes me laugh every time I do because it's such a middle school (?) thing to do but it works perfectly!

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michaelb2 made it! (author)2016-06-07

My first thought was....of course! My second thought was.....why didn't I know that! thanks

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smartrem made it! (author)2016-06-07

So clever and so obvious at the same time, I wish I had thought of this before. Love it!

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Charlie Kocourek made it! (author)2016-06-06

Yep, I usually start with paint cans but I like your method a LOT!!

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longwinters made it! (author)2016-06-06

It's like you know me when you mentioned the gathering of circular objects.
Thanks.

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seamster made it! (author)2016-06-06

This is a fantastic technique. Thank you for sharing how to do this!

author
MikeD50 made it! (author)MikeD502016-06-06

My pleasure :)

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Bio: I am a technical consultant in IT and Telecommunications, mainly for international data and multimedia networks. In my free time I enjoy sailing, being in ... More »
More by MikeD50:The last Splicing tool I buildConstruct curves to smoothly connect lines at any angleSimple marking gauge for non-square edges
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