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subject: Introduction On Intersection Of Curves: [print this page]

An intersection means that a single point that two curves or two lines meet or cross each other. Here the x + 2y = 3 and x + y =2 are the two curves intersect both at two points. Then it will never intersect anywhere. The line segment will exceeded in both directions and somewhere it should be parallel. Let us see about intersection of curves.

Intersection curve

Examples on Intersection of Curves :

Example 1

Find the intersection point of two curves

x - 2y = 4 and 2x + 2y = - 4

Solution:

Let the two curves be x - 2y = 4 -------------- (1 )

and 2x + 2y = - 4 -------------------- ( 2 )

Solving (1 ), we get

x = 4 + 2y

Sustitute the value of x in equation ( 2), we get

2 ( 4 + 2y ) + 2y = - 4

8 + 4y + 2y = -4

6y = - 4 - 8

y = -12/6

y = - 2

Substitute the value of y in equation ( 1 ), we get

x - 2 ( -2 ) = 4

x + 4 = 4

x = 4 - 4

x = 0

The intersection point of curves are x = 0, y= - 2.

Example 2:

Find the intersection point of curves 3x + 4y = 18 and 2x - 4y = - 8

Solution:

Let the two curves be 3x + 4y = 18 and 2x - 4y = - 8

Solving above two equations we get,

3x + 4y = 18

2x - 4y = - 8

--------------------------------

5x = 10 By adding above equations

x = 10/5 = 2

x = 2

3 ( 2 ) + 4y = 18

6 + 4y = 18

4y = 18 - 6 = 12

y = 12/4

y = 3

Therefore, the intersection point of curves are x = 2 and y = 3.

Introduction to special curves:

Let we learn about the special curves. Special Curves are commonly talking that an object alike to straight line. But they do not essential to be straight. In frequently curves that are in two-dimensional or three-dimensional. The word curve that has few meanings in non-mathematical language. They should have the meaning of through mathematical function that is learning curve or graph of a function that is Phillips curve.

Closed Special Curves

solve a math problem

This is one of the special curves. The parabola is the simple example of curve. The shape of parabola is shown below:

closed special curves

Closed curves:

Definition:

It is a curve it does not having the end points.

In this the starting point is joined always to the ending point.

Examples on Closed Curve:

closed curve

Related terms for closed curve are,

Curve

End point

Point

Find which one of the following figure is not closed curve?

not closed curve

Sol:

From the above figures,

The starting and ending point of figure 2, figure 3, figure 4 are joined.

But in figure 1, they are not joined.

So, figure 1 is the not the closed curve.

Answer: Figure 1

These are all the explanation about closed curve (special).

Types on Special Curves

Let we see about the two special curves:

Parametric curves:

It is a curve that can be used to done in the deformation of a straight line.

If should be the distance from some permanent point of the line.

That line is for each value of t and can specify the corresponding point over the plane.

Requirements of parametric curves:

It can be very easy to use for modeling. Therefore, it is used to making any preferred shape.

They are looking with smooth like the straight line

or piecewise smooth.

These parametric curves are drawn easily.

Bezier curves:

The Bezier curve can have the few following properties:

The endpoints of these curves are be interpolated.

Bezier curves tangents at endpoints are in the interior of control points.

The derivatives of this are 3(p1-p0) and 3(p3-p2).

This curve should be in the convex hull with the control points.

The formula for this curve is,

p(t) = (1-t)3 p0 + 3t(1-t)2 p1 + 3t2(1-t) p2 + t3 p3

Where, t is between 1 and 0.

by: johnharmer

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