You will need

- - a sheet of paper;
- pencil;
- a pair of compasses.

Instruction

1

Prepare in order to start the construction. To do this, draw two lines perpendicular to each other. The point at which they intersect, indicate the letter "O". This will be the future centre of the ellipse.

2

Define the essential values. The ellipse has the semimajor and semiminor axes. Before building label them a and b, respectively. As a rule, the length of these two segments is given in the problem statement to build an ellipse.

3

Take a compass and set the solution so that it was equal to the length of the segment a. Then set the compasses at O, and mark on one of the lines two points – P1 and P2. After that, the solution of the compass equal to segment b, designate two points on the second straight and name them Q1 and Q2. The resulting two line segments P1P2 and Q1Q2 are the large and small axes of the ellipse of the future, and the points – its vertices.

4

Find the foci of an ellipse. For this purpose a solution must be equal to the segment a. Set the compass at point Q1 or Q2, and indicate on the segment P1P2 two points F1 and F2.

5

Note on the segment P1P2 any point and call it T. Then, setting the compasses at this point, they measure the distance to P1, then draw a circle of this radius centered at the point F1. Next, draw another circle with a radius equal to the distance from point T to point P2, with center F2.

6

Mark the point of intersection of the two resulting circles. They belong to the desired ellipse. In order to draw the whole ellipse, you will have to repeat the steps described in the previous paragraph actions already new points arbitrarily marked on the segment P1P2.

7

Once you find a sufficient number of intersection points, connect them with solid line. This will be the desired ellipse.

Useful advice

To simplify and speed up the build process, instead of whole circles can be designated only the portion that is necessary to find points of intersection.

# Advice 2: How to draw an ellipse

**Ellipses**is called the geometric figure on the plane, which is defined формулойх2/a2 + y2/b2 = 1 a construction of an ellipse using compass and ruler, you must build points to her belonging.

Instruction

1

We introduce definitions related to the concept of an ellipse.

The two points F1 and F2 are called foci of the ellipse, if for any point M is taken on the ellipse, the sum of the distances F1M + F2M is constant.

The segment AB passing through the foci, the ends of which lie on the ellipse is called the semimajor axis.

Segment CD perpendicular to segment AB and passing through its middle is called the semiminor axis.

The two points F1 and F2 are called foci of the ellipse, if for any point M is taken on the ellipse, the sum of the distances F1M + F2M is constant.

The segment AB passing through the foci, the ends of which lie on the ellipse is called the semimajor axis.

Segment CD perpendicular to segment AB and passing through its middle is called the semiminor axis.

2

Let the length of the ellipse axes AB and CD. To construct an ellipse you can use the following algorithm.

We will have two perpendicular lines from the intersection points set aside segments horizontally is equal to AB/2 and vertically equal to CD/2

We will have two perpendicular lines from the intersection points set aside segments horizontally is equal to AB/2 and vertically equal to CD/2

3

Draw two circles with radius AB/2 and CD/2. From the center of the circle have a few rays.

4

Through the points of intersection of rays with constructed circles will hold the segments parallel to the axes of the ellipse.

5

Select the point of intersection of the constructed lines, it will be point belonging to the ellipse.

6

Connecting received points, will get an ellipse.

# Advice 3: How to find focus of ellipse

The shape of many real objects. For example, in the nature of an elliptical shape are the orbits of the planets of the Solar system, and in the art - sleeve. The properties of the ellipse resembles a circle and is its derivative.

Instruction

1

The ellipse is called the locus of points for which the sum of the distances of the two pre-specified points in the plane is constant. The shape of the ellipse is a flattened circle. He has so-called magic tricks, and for which the ellipse is constructed. One of its parameters is the focal length.

Before you draw an ellipse, read the definition of foci and their location. Mark the two focal points F1 and F2, and then spend a certain period of S. Draw an isosceles triangle so that the focal length F1F was his base. Point B is the top point of the triangle, and she needs to touch the arc of the ellipse.

Before you draw an ellipse, read the definition of foci and their location. Mark the two focal points F1 and F2, and then spend a certain period of S. Draw an isosceles triangle so that the focal length F1F was his base. Point B is the top point of the triangle, and she needs to touch the arc of the ellipse.

2

Once the triangle is constructed, make it mirrored, as shown in the figure, and draw the ellipse so that the line segment BB' is perpendicular to the segment F1F. Then the distance from point C to point F is called the semimajor axis of the ellipse and is denoted by the letter a. Twice the value of this axle 2A is equal to the segment S. Minor radius is the distance from the center of the ellipse to the point C.

3

Again, note the triangle CF1F. The midpoint of the segment On at the same time is the center of the ellipse and cut F1F, which, in turn, is the focal length of the shape. Note the triangle CОF and you will see that it is rectangular. And CF is the hypotenuse of the triangle, S is the smaller side OF the larger leg. To find the focal length of the ellipse, we need to determine the length OF cut. Known as the hypotenuse BF - the semi-major axis and shorter side S is the semiminor axis of the ellipse, then by the Pythagorean theorem way OF:

OF = √a^2-b^2.

Distance OF also sometimes called the eccentricity of the ellipse, denoted by the letter S. the Focal length calculate the following:

F1F2 = 2c = 2√a^2-b^2.

OF = √a^2-b^2.

Distance OF also sometimes called the eccentricity of the ellipse, denoted by the letter S. the Focal length calculate the following:

F1F2 = 2c = 2√a^2-b^2.

# Advice 4: How to draw an ellipse in isometric

**The ellipse**is an isometric projection of a circle. Oval build points and get on the patterns or curved lines. The easiest way to draw an ellipse in

**isometric**, inscribing a figure in a rhombus, otherwise isometric projection of a square.

You will need

- - the range;
- - gon;
- pencil;
- paper for sketching.

Instruction

1

Consider how to draw an ellipse in

**isometric view**, lying in a horizontal plane. Construct a perpendicular to the axis X and Y. the intersection of the mark O.2

From point O mark on the axes the segments is equal to the radius of the circle. Marked points indicate the numbers 1, 2, 3, 4. Through these points parallel to the guide axes of the straight.

3

From point O mark on the axes the segments is equal to the radius of the circle. Marked points indicate the numbers 1, 2, 3, 4. Through these points parallel to the guide axes of the straight.

4

Draw a arc from the vertex of obtuse angle of the connecting point 1 and 4. Similarly, connect the dots 2 and 3, having an arc from vertex D. Connect the points 1,2 and 3,4 of the centers of the small arcs. Thus constructed the ellipse in

**the isometric**inscribed in a rhombus.5

The second way to draw an ellipse in

**isometric**is to circle map distortion. Draw X and Y axis, from point O, draw two auxiliary circle. The diameter of the inner circle equals the minor axis of the ellipse, the major axis.6

In one quarter build auxiliary rays emanating from the center of the ellipse. The number of rays is arbitrary and the more, the more accurate the drawing. In our case it will be enough three auxiliary beams.

7

Will receive additional points of the ellipse. From the point of intersection of the beam with the smaller circle draw a horizontal line parallel to the X axis towards the outer circumference. From the top of the point lying at the intersection of the beam and of a large circle, drop a perpendicular.

8

The resulting point label figure 2. Repeat the operation for finding 3 and 4 points of the ellipse. Point 1 is located at the intersection of the Y-axis and the small circle, point 5 on the X-axis in place of the passage of the outer circumference.

9

Guide curve through these 5 points of the ellipse. At points 1 and 5 curve are strictly proportional to the axes. Spend a similar construction of an ellipse in

**isometric**on the remaining ¾ of the drawing.