In the former part, a space

curve, the parameters defining the curve

According to t, the solution that we have achieved curvature

twisting, finding a unit vector We learned.

Now this is what we have learned to consolidate as well as

To view an interesting problem, Let practices.

Proceeding to the application, our formula Let's get together.

Such expressions of b t n,

Expression of the expression of curvature and torsion like this.

As provided herein, now calculation method goes like this.

The first step, because the curve of x x We define with.

We need to calculate the first three derivatives.

point x x2 x3 point point because, at t x point b

There are points x2, x2 points off there and in to, there x3 point.

Then, after the calculation of their these multiplications 2s

3-multiplying and multiplying these lengths account We have to this because they

formula is set in the third step in -putting them.

Now that's a spiral helix Let's examination,

ie before the construction of a helical Let parametric equations.

Maybe some of you have done, yapmadınızsa can be done easily.

As on a page,

draw a line of a circular shape that

I Bükel on a circle that is, a on a circular cylinder.

, On which the straight line drawn, us will give a helical spiral.

If we look a bit more digital, horizontal 2 pi times

a, b vertical length of 2 times pi If we choose, horizontal

we translate this into a circle is the radius of a circle wall.

This slope also seems to be a divided by b.

With these thoughts helix equation, x

A times the cosine t happen, as it on which we

The projection of this point moves falls on this circle.

On this circle is equal to theta If we think we're starting from scratch,

It is a times cosine theta x, y at a time is sine theta.

Parametric equation of a circle, but The difference between the circle, the circle always in the plane

Standing here on the way back in, b

in line with the rise of this the helix formed this happens.

Thus defining the helical vector here

Or seen as a cosine of theta t, We chose here as a parameter.

be a sine t and b t.

As we define the spiral

Let's just its derivatives.

immediately by the point x such that x from derivatives

t minus sine and cosine t come b.

When we take the second derivative of x points We take one more derivative.

I also point x3, x2 point again We know that with derivatives.

Now, in this formula, we point x and x2 noktanınvekt needed to multiplication.

The length of point x, the point x of the point x2 multiplying the length of a 3-in.

Now if we make them, with point x

vector product of the point x2 work determinants comes from.

i j k are writing.

Used as the first vector point x we put on line.

x2 point we put in the last row and We are opening.

I components of the first row and column tumbling these

zero, but the pros will become negative, the second

diagonal change of sign changes Due to changes in the EU once the sine t.

.mu.l we're looking at, minus the EU in a similar manner times the cosine cosine t involved.

K, .delta we're looking at here is an interesting There formation.

See a squared sine-squared t, where a square For frame t is the cosine,

staying here is just a square, so this expressed very simply appeared.

If we multiply this by taking away the point x3 See here x3 point

We found point x x2 point.

It is also very interesting from the inner product simplification going on, see

EU sine-squared t, b times a here,

cosine squared t, then the two combination

b involved because a square sine-squared

t is 1 to t plus cosine square, a is multiplied by the square zero

For one thing does not come from there, see How easy was this triple product.

When we look at the length of point x, x

point was here, see the first one here frame, plus

second squared, sine-squared t plus for giving a squared cosine squared t 1,

third-squared plus b squared here, plus b squared is coming from Accra to say that,

If we define it as the definition of c frame, x is a fixed point of the square.

Gene vectors with point x x2 point

Taking the length of the product, see here again, sine-squared t,

kosinüzs frame t is happening here, ab, a square b squared is coming.

Here comes again than a square, the accounts here

You can easily see, the AC turns out, this Coming up with a simple structure.

When we put them to behold If we rewrite this

We found the size, t, by definition, or the formula

x at point x on the length of the point It was part of the x point

We found that a certain length c, divided by the it turns out.

because we need to point t n, t point t point

longitudinal division, t, t certain point now easier to find, Sinisa will be cosine,

will be minus two is sine cosine out to be negative for the negative

turns out to be constant b is divided by c to b its derivative zero.

So we did not find t point point their

divide here by dividing the length of the c going.

Already it is one of the neck

easily see, cosine squared plus sinus

square point x and x2 port vector b multiplication, we know that in line,

the unit vector divided by its length This work will be of this size here

We take the size of its longitudinal split time

As you can see a time it turns out to b.

Now here are a variety of delivery possible.

The inner product and the inner product of t

is zero because it is orthogonal need.

See here sine cosine t t here t sine cosine t, different signs

At the same coefficients, where the reset This is because the product is zero.

Again let's look at the product of n and b.

Sine cosine, sine and cosine again

different sign, and the third is zero, again output from the zero.

If you hit it in a similar way to zero in We see that go.

There were as curvature and torsion,

point x of the point multiplied by x2 length, it

We have calculated the point where AC x

c divided by the cube of the length of the point x respectively.

You can see from the division of c cubes As we come to this conclusion.

And finally

iii of this triple product of these two product longitudinal

are divided here b divided by c square is çıky.

Helical structure to emphasize the privileged because I got

output constant curvature, torsion also constant output,

How the curvature at each point on the circle

If that is going hard at it on the helix on spiral

at every point of the curvature and torsion constant involved.

that does not change from point to point.

If we reverse the problem, curvature and found torsion is constant curve

one has to say a huge curve in the universe, both curvature and torsion is constant.

Now comes to mind right now, this spiral, 've read the double helix

Perhaps you've heard at least, consists of a helix structure of the DNA

I wonder where I wonder if there is a relationship molecules that together form

Seems curvature and torsion

A constant travels around the subject our work

Or something out of nothing in excess, but The interesting thing is certain.