Here are 2 more ways to draw arcs that are accurate and that will fit a predetermined dimension. One of these methods allows you to use a predetermined arc radius.
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Arcs are very common in home construction, furniture design, sign making, and architecture. Usually, the arc needs to fit in a space of predetermined width or height. Sometimes, but not as often, the radius of the arc needs to be determined or used. If a large radius is used, it becomes awkward to actually use the radius due to the amount of room needed. In these cases, it’s better to use (somewhat) simple algebraic formulas to determine the layout.
This episode will show methods 3 and 4 for drawing an arc of a predetermined height and width. These dimensions are more commonly known than the radius of the arc, or the size of the circle that the arc is being “sliced” off of.
You can view Episode One where DirtFarmerJay shows you how to draw a fair arc using a flexible piece of material and nails for one method, and nails and a tapered board for the second method.
Method 3 can be used when you know both the height of the arc, as well as the baseline width. Using paper, 5 nails, 2 wooden yardsticks/lath pieces, and a pencil, you can easily draw a fair arc.
Method 4 is used when you know only two of the three dimensions of an arc: 1.) the width (baseline) of the arc, 2.) the highest point of the arc from the center of baseline (known properly as the sagitta), and 3.) the radius of the arc. In all fairness, this is a “somewhat” simple way. It takes calculating, using an algebraic formula.
Frankly, this DirtFarmer ain’t smart enough to figure these things out on his own. I reached out to www.freemathhelp.com and asked about how to figure this out. Dr. Peterson responded almost immediately with authoritative information and the link provided below. Thanks, Dr. Peterson, for pointing me (and ultimately our viewers) in the right direction. YOU ROCK.
The link Dr. Peterson supplied offers three different formulae, the one you choose depends on what dimension you are missing.
For instance, if you know the height (sagitta) of your arc, as well as the baseline, but want to determine the radius, one method will work. So it goes for each calculation. If you know two dimensions, but are missing the third, you can calculate it with the appropriate formula.
Hey, Dirt Farmer Jay here from DirtFarmerJay.com.
In a previous video,
I showed you how to draw a perfect arc.
I've got two more great methods you're gonna wanna see.
Stay tuned.
Hey, Dirt Farmer Jay here from DirtFarmerJay.com.
Welcome to part two of how to draw a perfect arc.
In that first episode,
which you can check out right here by following this link,
we showed you two methods for drawing
what's called a fair arc.
In this episode, I'm gonna show you two more methods.
One that's just kind of a cool trick,
kind of quick and dirty,
and the other one's super geeky for those of you
that have gotta get something a specific size and radius.
Before we get started,
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Well, let's get to work.
I'm gonna show you two more methods.
All right, the first method I'm gonna show you today
is when you have two pieces of information.
You know how wide the arc is.
It's gonna fit in a predetermined area.
And you know from that baseline how tall the arc is.
If you have those two pieces of information,
then follow along on this method.
And you can go back to that first video
for two other methods
if you have just those two pieces of information,
width and height.
Well, let's go ahead and do this.
We're gonna go ahead and draw a baseline here.
We've just done some real simple things.
All you need is a straight edge.
You need a square.
You need some nails.
And in this case, you're gonna need five nails.
Of course, a hammer to drive them.
You need two pieces of wood.
This case, I got some inexpensive wood rulers
or yardsticks, but pieces of lath will work just as well.
You'll need some brads and a hammer
to put those brads together.
We're gonna make an arch that's 34 inches wide
from point to point.
That's in geometry terms
or in formal terms,
that's called the arch's baseline or cord.
So that's what we're gonna do
is we're gonna do a 34 inch cord.
So let's just get that drawn right away here.
And I'm just gonna draw a wild line,
starting at 34.
And we're gonna go all the way down here,
down to zero.
And now we have a 34 inch baseline.
The second thing we need to do is bisect that baseline
and find a center point.
So half a 34 is 17.
So I'm just gonna put the 17 inch mark right there.
And just to double check, we know it's 17.
I'm gonna slide it there and roll it down to here.
And you can see we got 17 on both sides.
So now we have a baseline with a center point.
Now we're gonna draw the height.
And I'm just gonna put a T here.
Now, as I told you,
the formal name for this or the correct descriptor
is called a cord or baseline.
This height right here is not just the height,
it's known as a Sagitta.
Does that sound familiar, Sagittarius?
Has to do with an archer, an arrow, a point.
So what we're gonna do here is let's just choose a number.
Let's just do a seven inch.
So I'm gonna run this up
and I'm gonna stop right at seven inch.
And there is my Sagitta or the height of the arch.
So when we're done here with this technique,
you're gonna have an arch that originates
right at that point.
It's gonna be a fair curve.
In other words, it's gonna be beautifully smooth.
There's no flat spots.
It'll be this beautiful curve
that at its highest point hits there
and then does the exact thing.
It regresses back over into this point
and those two sides are equal.
There's no football shape or anything like that.
It's just beautiful.
We have those three points.
Now, what do we need to do?
Well, we actually need to pound some nails,
but we need to do one more thing for this method.
I need to get this point moved up
to the same height over here and this one over here.
So I'm actually gonna draw a parallel line here.
Let's do that.
All right, now we need that point over here.
We know it's 17 inches,
so I'm just gonna measure 17 inches
on both sides of the center point and put a mark there.
Gonna do the same thing over here, 17 inches.
And I've got four points now.
Let's drive some nails.
I don't need a nail here, but I need a nail right here,
right here, right here, right here, right here, five nails.
Let's go ahead and drive just enough
into the top to be solid.
Here we go.
All right, we now have our points all put together here.
Now, this is where the magic happens.
What we're gonna do is take these two straight edges.
And again, these aren't super precise,
but it's probably close enough for most shop work
and especially smaller arches.
Again, I want you to notice
that this could be scaled up, a longer arch,
let's say for a doorway transition
between two living areas in your home,
something that might be used as an outdoor hardscape item
out in your yard scape,
and you want this beautiful, fair arch,
you could scale this up.
And as long as you lengthen the sticks,
and you'll see why in just a moment,
you can use this same technique.
So the first thing we're gonna do
is lay this against two of the nails,
and it doesn't matter whether you do it these two
or that two, but we'll just go ahead and start right here.
And you want that to overlap here.
Then you're gonna take this one
and lay it over the top of,
butt it up against this nail and against that nail.
And then what we're gonna do here,
I'm just gonna pretty it up a bit
as I'm gonna drive some fasteners right here
to affix this angle.
So let's go ahead and do that.
We're gonna draw the left side,
your right side of the arc to begin with.
And so we'll lay the top stick along the outside edges
of these two nails,
and the angle stick on the outside edge
of this outline nail on the baseline.
Now, right at the intersection of where these two happen,
I'm gonna drop the pencil point in here.
And while there's a little bit of variation,
you're able to clean it up afterwards.
We're simply gonna put it here,
nest the pencil in here,
and then we're gonna move this,
keeping two of the nails in contact at all times.
You can't do this and you can't do that.
If you do, you've lost it.
So the idea is to move in a nice smooth action like that,
keeping the pencil in here.
Well, let's see what it does.
Here we go.
You have two choices here now.
Either I can take this and fold, undo it,
fold it in half and transcribe that line over here
to get the other side,
or I can now reverse sides and put this here.
Same thing.
Sitting on the outside of these two nails
and on top of this one,
we're gonna put the pencil right in the corner here,
and we're gonna move to the right.
What do you think?
That's a fair curve.
Now, you should notice that this is actually
a slice of a larger circle.
So if the circle would probably be
just about this big around,
and what we've done is done a tangent,
just cut off a section of it.
Again, this is great,
along with the other two methods we showed you
when you already know the baseline length
and the sagittal or the height.
What if you've got a predetermined radius?
That is, let's suppose it's in a large area
or a big display area or maybe a showroom or something,
and you want the same arch to appear and disappear
off a center radius,
and you have a predetermined radius.
Let's suppose that the radius of the circle
in which you're cutting off tangents is predetermined.
Let's just choose something.
Let's say it's 25-foot radius, a 50-foot round circle.
Well, you could imagine trying to take a radius,
a 25-foot radius, find a space big enough
that you'd create kind of like a compass
and then draw an arc and slice a piece off of it.
That's not the way to go.
So, geek alert.
The fourth method takes some algebra and geometry,
and it is based on formulas.
Given two of the three dimensions,
you can always get the third one.
So, if you know the radius and the baseline,
one formula that we're sending you to
will give you the sagittal.
If you already have the sagittal and you have the radius,
then another formula will give you the baseline.
So, two givens and then one thing you find.
Since you can usually draw an arc
using one of the first three methods
if you have the baseline or the chord
and the height here of the sagittal,
the one that really may be helpful for you in the formulas
is if you already have the radius
and how wide the baseline needs to be,
what do you need as far as the sagittal?
You can follow over at the websites
that we've provided below in the description,
and you can check out those websites
that'll walk you through algebra.
I'm not gonna teach any algebra or geometry on this channel,
but for those of you that really need those tools,
you can go right over there
and you'll get the help you need.
And speaking of not using math,
check out this video episode
where I teach you how to find the dead center of any board
without having to use math.
And while you're at it,
check out this video that YouTube thinks is interesting to you.
Well, that's not bad.
