You will need

- - the point with coordinates (a,b,c);
- - coordinate system.

Instruction

1

*Draw the three coordinate plane to have the origin at the point O. In the drawing, the plane of projection are designated by three axes Oh, Oh and oz, the axis oz directed upwards, the axis of the shelter to the right. To build the latest ox-axis, divide the angle between the axes OU and oz in half (if you draw on paper, just slide the axis on the diagonal cells).*

2

Please note, if the coordinates of point A are written as three numbers in brackets (a, b, C), then the first number a is the distance from the plane x, the second by b – have, a third c – z. First, take the first coordinate and mark it on the ox axis, to the left and down if the number is positive, right and up, if it is negative. Received the letter V. name

3

Next, take b and place it on the axis Oy – the right positive and left if negative. Name the marked

**point**with the letter S.4

Then put the last number with the up axis oz, if it is positive, and down on the same axis, if negative. Mark received

**a dot**letter D.5

From the points guide the traces of the projections of the required point on the planes. That is, at point b, draw two straight lines which are parallel to the axes OU and oz, at the point C draw straight, parallel to the axes ox and oz, at the point D is straight, parallel to Oh, and Oh.

6

Two straight, held in the same plane, intersect. Restore the normal (all three planes) to find the desired

**point**. As a result, you will receive a drawing of the box, mark the**point**with the letter A. Check the distance to the planes this point is equal to a, b, C.7

If one of the coordinates equal to zero, then the point lies in one of planes of projections. In this case, just check the known coordinates of the plane and find the

**point**of intersection of their projections. Be careful when you build the points with**coordinates**(a, 0) and (a, b, 0), do not forget that the projection on the axis ox is at an angle of 45⁰.# Advice 2 : How to find a point on the line

In modern mathematics a point are called elements of very different nature, composed of different spaces. For example, in n-dimensional Euclidean space a point is called an ordered set of n numbers.

You will need

- Knowledge of math.

Instruction

1

Video - one of the basic concepts in mathematics. Video analytic on the plane is given by the first order equation of the form Ax+By=C. the Identity of points to a given

**direct**easily determine by substituting the point coordinates into the equation**of the straight**. If the equation is drawn in true equality, then the point belongs to**a straight**. For example, consider**a point**with coordinates A(4, 5) and the straight line given by the equation 4x+3y=1. Substitute in the equation**the direct**coordinates of the point A and get the following: 4*4+3*5 = 1 or 31 = 1. Got equality, which is not true, and therefore, this point does not belong to**direct**.2

To locate a point on

**a straight**is enough to take one of the coordinates and substitute into the equation and then Express the resulting equation from the second. Thus there is a point with a given one of the coordinates. Since the line passes through the entire plane, then the points that belong to it are infinitely many, and hence for any one of the coordinates will always be another, such that the resulting point will belong to a given**straight**. Take for example the line with equation 3x-2y=2. And take the-coordinate is x=0. Then substitute the value of x in the equation**direct**and receive the following: 3*0-2Y=2 or y=-1. Thus we found**the point**lying on the**straight**and its coordinates are (0, -1). Similarly, it is possible to find**a point**belonging to**the straight**, when we know the y coordinate.3

In three-dimensional space a point has 3 coordinates, and the line is given by a system of two linear equations of the form Ax+By+Cz=D. in a Similar manner as in the two-dimensional case, if you know at least one coordinate of a point solving the system will find the other two and this point will belong to the source

**direct**.Note

After finding all the point coordinates, you must verify their correctness. Substitute the found coordinates into the equation of the line and if will be true equality, all solved correctly.

Useful advice

A way of finding a point in the known coordinate are valid for any space dimension, the only difference is how many steps are needed to solve equations, to find the other coordinate.

# Advice 3 : How to find the projection of a straight line on the plane

Projection of one or another volume of the object is called its image plane. The ability to build projections needed for the representatives of different professions. It's so common that people often even do not think about it, just make plans and maps, pictures of the items under a particular angle, etc. First of all, you need to learn how to build the projection of a straight line.

You will need

- - plane;
- - video does not match the given plane;
- - gon;
- pencil.

Instruction

1

Imagine a plane on which to project direct. Label it, for example, on a sheet of paper. In any case, you get the image plane, so it will be rather conditional. You have to apply some spatial imagination to imagine how it will look in three-dimensional space.

2

Swipe a line that does not lie on this plane. Of course, you can build the projection of any line belonging to the plane, but it will be identical to the original line. If no additional parameters are specified, construct an arbitrary line. But in the conditions can be specified and the angle between your objects. It is formed at their point of intersection.

3

In order to build the most simple orthogonal (rectangular) projection, lower of any two points belonging to a given straight line, the perpendiculars to the plane. Connect them with a segment. This will be a rectangular projection of this line onto the plane. If the source video is a segment, then it is necessary to project its ends. For the beam is taken as the starting point and any other. For direct of arbitrary length it is best to first find the place of its intersection with the plane.

4

For better understanding, formed as a rectangular projection, do the following experiment. Hang on the wall a small screen. You can use just the white wall. You will need another piece of rope and 2 chairs or attached to the Desk stand as well as any light source. Tie a rope to the uprights. Place them so that the cord was stretched and at an angle to the screen. Measure the cord.

Place a table lamp so that its beam fall on the screen at the right angle. It needs to cover and the rope. Paying off the overhead light, you will see that the tense cord casts a shadow on the screen, and the length of the shadow does not coincide with the length of the cord. The shadow is the projection — in this case, rectangular. Experiment by placing the light source so that its beam fall on the screen at different angles.

Place a table lamp so that its beam fall on the screen at the right angle. It needs to cover and the rope. Paying off the overhead light, you will see that the tense cord casts a shadow on the screen, and the length of the shadow does not coincide with the length of the cord. The shadow is the projection — in this case, rectangular. Experiment by placing the light source so that its beam fall on the screen at different angles.

# Advice 4 : How to build a point with three coordinates

If descriptive geometry is the theoretical Foundation drawing, build a point in space on coordinate – based geometry. The position of any point in space, you can specify three

**coordinates**, and, if you have three planes of projections, you will not be difficult to find her.You will need

- - the point with coordinates (a,b,c);
- - coordinate system.

Instruction

1

*Draw the three coordinate plane to have the origin at the point O. In the drawing, the plane of projection are designated by three axes Oh, Oh and oz, the axis oz directed upwards, the axis of the shelter to the right. To build the latest ox-axis, divide the angle between the axes OU and oz in half (if you draw on paper, just slide the axis on the diagonal cells).*

2

Please note, if the coordinates of point A are written as three numbers in brackets (a, b, C), then the first number a is the distance from the plane x, the second by b – have, a third c – z. First, take the first coordinate and mark it on the ox axis, to the left and down if the number is positive, right and up, if it is negative. Received the letter V. name

3

Next, take b and place it on the axis Oy – the right positive and left if negative. Name the marked

**point**with the letter S.4

Then put the last number with the up axis oz, if it is positive, and down on the same axis, if negative. Mark received

**a dot**letter D.5

From the points guide the traces of the projections of the required point on the planes. That is, at point b, draw two straight lines which are parallel to the axes OU and oz, at the point C draw straight, parallel to the axes ox and oz, at the point D is straight, parallel to Oh, and Oh.

6

Two straight, held in the same plane, intersect. Restore the normal (all three planes) to find the desired

**point**. As a result, you will receive a drawing of the box, mark the**point**with the letter A. Check the distance to the planes this point is equal to a, b, C.7

If one of the coordinates equal to zero, then the point lies in one of planes of projections. In this case, just check the known coordinates of the plane and find the

**point**of intersection of their projections. Be careful when you build the points with**coordinates**(a, 0) and (a, b, 0), do not forget that the projection on the axis ox is at an angle of 45⁰.