Hello. Bi-plane curvature in our previous session I began to recall the concept. Because it before moving to the curves in space would be useful to know. Now the geometric mean radius of curvature I want to give you some information. The inverse of the radius of curvature mean curvature mean. It has a nice geometric meaning. Already this curvature, the tangent of the slope, ti to the length of plug extension, derivative is defined as that the radius of curvature of this curve is drawn tangent to the curve plotted as have to do with the bi circle. B has relations. And this is one of the accepted definition is consistent with our intuition to see need. Let's start with a question like this. What is the curvature of a line? It's like a little word game. While it is true does not curve curve B is correct. We say the right to correct curvature should not be. Let's see if this foreboding consistent? In general, an accurate equation of x a first force and y comprising function. y is equal to x plus b. A fixed first derivative thereof used derivative also zero. E, it took curvature equation When we place you can see the denominator is zero as the base year for two curvature zero. This foreboding consistent? Yes. Yes. Consistent. Because e, b of the curvature is not correctly In need of definition. Here's the really zero curvature We find that the numerically. This is in accordance with our intuition. Let's do an example of the second b. Consistent with our intuition curvature circle Bi Do you? The answer is yes. What would hunch about us? No one interested in mathematics If you h b How do you identify the curvature of the circle? As a bi yarıçapıl small hoops such a large circle of radius b. Larger semi-circle b. This is immediately seen people in the smaller the radius of curvature increases. Rotating much more quickly. Radius grows faster than you can go curve does not return. Let's see if you will give it? Explicit function of the equation of a circle Such as we show. Because this is equal to x squared plus y squared obtained from the square. Top job when he calculated its derivatives force coming forward. As one-half. We take the derivative of the inside. There's always chained derivatives rules is there. Minus two for the derivative of x is coming. The front of this force falls by two Two one-half each other brings. 're Getting this size. Our curvature plus years in the base formula We need to frame. One plus years we calculate the square base. One of the x squared plus y squared divided by base There denominator squared minus x squared. In the denominator. In the same denominator as you can see when you bring a because one squared minus x squared is coming. Here are the plus x squared. x squares refinement. As you can see, see a very simple size're getting. This year we found two base year of base Taking the derivative means once again. This year, when you get there it is a derivative of the base has a multiplication. As times have u. First hit the second plus the derivative the second derivative of the first times. This account also try y at the two base With such a simplification of b, s, that Coming structure. Also put off by this as a negative difference but for now it will be more trouble to do. Retrieving because the square root of a lot of blah. This more precisely how the görecez will be made. This is a plus y squared denominator of base We're putting. Taking the square root of the cube root, ku, cube, third force cube when you receive an email here, a cube is made denominator in this, ka, divided by the denominator of the three second force is going. two years in a row already has a squared bi. Bi three-way split of the second force again is there. This term X, simplifies each other Coming to a trailing back and forth. If a curvature radius of curvature in a divided Conversely, it is a. This is consistent with our intuition? Yes. Because greater than the radius of a circle, a Close grows smaller. That is, the curvature decreases. So this is a reasonable thing. If a radius of a tiny much more quickly You are returned around. Hence this is faster and Bending with a radius of arrive exactly equal to the radius of curvature an appropriate conclusion to our intuition. Doyı at the beginning, this is divided by d f h derivative, receipt of this measure on this önsezgi give results that are appropriate grounds. Now I am giving an example here. But it will pass quickly it alone I would advise you to do. He let a parabola. y equals x squared one-half. X is equal to zero, the curvature thereof, we saw that in the circle of curvature at point x independent, but in the other curve curvature depends on where we are. If you are already following question:reverse h constant curvature What is the curve so that the interests of a single circle. There is no other curvature of the curve is constant. Thus, X, with a parabola will change curvature. We want this one in x is equal to zero. So this is like a parabola geliyo. We want it's in your lowest point. We're Here 'x y Base account. We are two home base year calculation. A derivative of x. y when x is equal to zero, we have the account base becomes zero. Put them by taking away two base years a denominator. Zero in the denominator of this year, the base year is zero three for one split second force. He is also a course. We find here, and here's a close curvature Conversely, it is also a trap radius involved. Now let a question like this. So this parabola tangent circles Let's draw. Draw a large circle of radius b of the When you see In these two points will be cut as two parabola. Here the condition that the tangent we want. This is the equation of the circle of curvature now something independently center y is equal within a radius which is also a This is the equation of the circle. x squared minus a squared plus y minus is coming. As seen here this year opened minus The square frame of a future year. Less come twice a year. BI will also squared. But right there in the bi squared. They's led them to zero back and forth to each other to further refine this will remain. Now we have this circle with the equation of this parabola We find cut points and it is very easy. If we put y instead of x squared divided by the following we obtain. As you can see here x squared common is a multiplier. If we take x squared here comes one. Less income here as well. A negative. Here's comes the x squared divided by four. Here right into the water, can be seen. If you see a minus here is greater than a value will occur. Therefore x squared divided by four is a plus will be worth it. Let's watch this here. If greater than that of three intersection point will be. One of them is already zero x squared radical. One greater than the old one is the square root of a real number in will be released. But if a small one where the brackets b of the root does not. Has only x is equal to zero. Where a is equal to the See Also decreases. Yet a square root of x occurs. From inside the brackets of four, two-storey here here is a two-storey four-storey zero happens. So the equation becomes x four. This shows the following. when there is less than one and a tangent does not intersect but that is tangent to the parabola of the cut-off The largest of the circle When we say the E, ie an equal a radius of the circle is going on. Also. If we compare the radius of curvature of these To see the meaning of curvature. Only the biggest but the curve radius tangent Another point that touches the cut in going in a circle. Is the radius of the circle is the largest of the curve meaning in the overall picture of him in we can see. This is a general curve of a circle to a curve Let's draw. Here we see too big radius from here they will cut, but Only by keeping a smaller radius tangent which is able to find. But the greatest of these infinitely many circles radius to the radius of curvature circle that is going on here details you can see. To understand this thoroughly over I would advise you to switch. The circle is the only curve constant curvature. Therefore, money, independent of x, circle curvature, regardless of where we are. When you change the x point of the parabola curvature will also be changed. Here I give you as an assignment, road show, in particular the previous step, You will follow the steps. I gave the answers in your answer here Be able to check the response that you get, I look AmAsInI provides. We said at the beginning usually in space but more useful parametric in plane Parametric representation to be advantageous I have seen situations. For this reason, curvature of the parametric representation There are also benefits in terms of our acquisition. Indeed, the parametric equation of circle representation is easy. We used the following shorthand notation. x on the derivative with respect to t How short the dot as the base year If doing impressions derivative of y with respect to t again at this dot This very derivative components of the vector they will be. Here the second derivative and curvature formula first derivative needed. So let the derivative of y with respect to x but the function y x it is not given as to We can not calculate, he accounts directly We can not calculate for him, chained, Thank indirect derivative calculations. y with respect to t, t by the x We take the derivative. This means the following. y point denominator, d t d x, d x divided by split d t because it comes in the denominator. In this manner parametric variables y base denominated finds. Y do we calculate the square of the base is a plus. Then two years, our home base to account You have to again in year two base because we can not directly account base x of y not provided as a function which t given function. Here for the first t based on indirect calculations derivatives with respect to x then the derivative of t we get. Where d t fractions but symbolic When we look at Even this simplified d t d y this is it base is divided by x stayed back. But of course this is not the fractions can not be simplified is a suitable choice of illustration only To determine telling. Taking the derivative with respect to t, where y prime We work to find you here again, what base year is. Derivative with respect to t, we can get it anymore. Put this account of whether, where open supplied with representation, ie, y f x as The formula given by the curve parametric shown that x t and y t in representation we obtain the equivalent formula. Now here's to a natural cut-off point We came. We need to keep a little more on plane curves. Because when we do not understand it well better to understand curves in space We'll enforced. They pass through a bit of thoroughly I'm waiting for you to learn. Then we can continue. Goodbye.