# [Descriptive geometry] Scan cylinder intersecting with a cone

- Data to build a cylinder sweep
- How to make a cylinder scan (algorithm)
- How to do a base scan straight cylinder
- Construction of the line of intersection of a straight cylinder with a cone on the scan of the surface of the cylinder

[Descriptive geometry] Scan cylinder intersecting with a cone

## Data to build a cylinder sweep

Given: __ The intersection of the cone and cylinder __ - two intersecting surfaces - the surface of a straight cone and a cylinder - the line of their intersection.

Necessary: **Make a scan of the cylinder and put on it the line of their intersection** .

In the previous video tutorial " __ Cone reamer __ "we built an approximate sweep of the cone, inscribing a regular 12-sided pyramid into the cone. We **also do a** rough **construction of the cylinder sweep** , dividing the base of the cylinder into 12 parts.

## How to make a cylinder scan (algorithm)

- We build the scan side surface of the cylinder.
- Divide the base of the cylinder into 12 equal parts.
- We measure the chord between any two adjacent dividing points of the base circumference and put this distance on the bottom side of the cylinder sweep.

- Attach the base of the cylinder to any forming side surface.
- Draw on the scan of the side surface of the cylinder the line of intersection of the cone and the cylinder.

## Construction of the lateral surface of the cylinder

The sweep of the side surface of a straight cylinder is a rectangle. The height of the rectangle is equal to the height of the cylinder, and its length is equal to the length of the circumference of the base.

Since we need to build a scan of the side surface of a cylinder intersecting with a cone, this scan will be a rectangle with cutouts.

We will use a simplified method for the development of a lateral surface of a cylinder. To do this, we divide its base into 12 equal parts on the frontal plane of the projection. It can be done, __ inscribing a regular polygon in a circle __ .

Note __ characteristic points __ the intersection of the vertices of a dodecagon with a circle. In the figure, these points are marked with serifs and crosses. We connect these points with segments - chords.

Draw the side surface of the cylinder, taking the height of the rectangle from the horizontal plane of the cylinder projection. Take the length of the rectangle from the frontal plane of the projection of the cylinder, which will be equal to the length of 12 chords. Measure the length of any chord and set it aside 12 times on the surface of the cylinder.

## How to do a base scan straight cylinder

The base of the straight cylinder is a circle. The cylinder has two bases: upper and lower. Therefore, take the radius of the base of the cylinder on its frontal or horizontal projection plane and attach it to the lateral scan of the cylinder, as shown in the figure.

## Construction of the line of intersection of a straight cylinder with a cone on the scan of the surface of the cylinder

Since the generators of the cylinder are projected in full size on the horizontal plane of the projection (the cylinder is frontally projecting), we will take on it the coordinates of the points from its base to the intersection line and transfer it to the scan of the cylinder in order. The numbering of the cylinder generators is shown in the figure (you can choose for yourself any sequence of removing the coordinates of their intersection line for the cylinder sweep, the main thing is not to get confused).

I will show the principle of removing the coordinates of several points of the line of their mutual intersection and drawing them on the lateral scan of the cylinder, the rest will be done in a similar way.

You already have 12 cylinder generators, which you got when building its lateral sweep, and in fact you can rely on them in order not to get entangled in the cylinder generators. **Also, there are already characteristic points of the line of their intersection on the frontal and horizontal projection plane, you only need to transfer them to the lateral scan of the cylinder surface!**

Determine the first extreme reference point of the intersection of the cylinder with a cone. It is located at a certain distance from the 3 generators of the cylinder, which you can measure on the frontal plane of the projection and set aside from the 3 generators on the scan of the cylinder. Further on its development, draw an additional generator and postpone this point. This point lies on the axis of symmetry of the side surface of the cylinder, the middle of the generator.

You can build an additional line on the side surface of the cylinder, which will divide it in half. This is necessary for the convenience of putting the coordinates of the symmetric points of the line of intersection of the cylinder with the cone on the sweep, so as not to postpone each point from the base of the cylinder, and from the "axis of symmetry."

Apply the following points of their line of intersection on the lateral ramification of the cylinder. They lie at a certain distance from the 4 generators of the cylinder and are projected onto the frontal projection plane at a single point, since the cylinder is forontically projecting. Measure this distance and set it aside on a 4 cylinder sweep of the cylinder. Spend auxiliary generator on the scan. Now on this generator of the cylinder it is necessary to put down the points of the line of their intersection whose coordinates you can take on the horizontal plane of the projection.

Thus, you transfer all points of the line of intersection of the cylinder and the cone to the scan of the surface of the cylinder and connect them with a smooth line.

**In the video tutorial, the principle of constructing the development of a straight cylinder and drawing a line of intersection of a cylinder and a cone on it is shown in great detail and clearly, I recommend for viewing.**