**SURFACES of REVOLUTION**** **

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Surface of revolution cam be created by revolving a curve segment about the given axis. Axis of revolution *o* is an arbitrary line in the space. Geometric transformations (revolution and transaltion) can be used to locate the line *o* to one of the coordinate axes, or to the position parallel to one coordinate axis. In the Fig. 4. 41 the line *o* is translated in the direction of vector **a**=*PO* to the line *o*´ passing through the origin of the coordinate system, then it is revolved by angle a ^{x} aboot the coordinate axis *x* to the line *o*´´ and at last it is revolved about the coordinate axis *y * by the angle a ^{y} to the cooridnate axis *z*. Matrix of the revolution about the arbitrary line *o* is the product of the multiplication of matrices representing separate geometric transformations, matrix of translation and matrices of revolutions about the related coordinate axes *x*, *y*, *z.* In the orthographic views of the surface of revolution it is covinient to locate the axis of revolution to the line parallel to the coordiante axis *z* (in the Monge method) or directly to the coordinate axis *z* (in the orthogonal axonometry), Fig. 4. 43.

Synthetic representation: (*k*, **T**_{Oz}(*v*))

Analytic representations:

- basic figure -

- generating principle - class of revolutions about the axis

- modelled figure

- (

Surface of revolution with the axis in the axis *x*, which is created by revolving the Archimedean spiral

by angles from the interval <0, p> is illustrated in the Fig. 4. 42.

**1. Parametric net of curves on the surface of revolution**

Net of parametric curves on the surface of revolution

is created by *u*-curves congruent to the basic curve *k*

(separate position of the basic curve *k* in the revolving about the axis *o*)

and *v*-curves, trajectories of the points on the basic curve *k*,

parallel circles (located in planes perpendicular to the axis *o*).

In any point *P* on the surface

there is located one parallel circle *l ^{P}*

and one revolved position of the basic curve

Plane m passing through the axis of the surface of revolution is called the meridian plane. It is the plane of symmetry of the surface. Any surface of revolution has the infinite number of planes of symmetry, any patch of the surface of revolution at least one. Planar intersection of the surface of revolution and plane m is called

(*k*, **T**_{Oz}(*v*))=(*m*, **T**_{Oz}(*v*))

In any point *P* on the surface there is located one parallel circle *P*Î*l ^{P}* and one meridian

Tangent plane in the point

Meridian plane parallel to the frontal image plane is called principle meridian plane

and intersection of the surface of revolution by this plane is called

Tangent * ^{m}t* in the point

Vertex *V* of the tangent conical surface is located on the axis of revolution.

Normals to the surface of revolution in points located on one parallel circle *l*^{ } are located on the normal conical surface of revolution with the axis in the axis of revolution and vertex *W* on the axis of revolution, while lines on this surface are perpendicular to the lines on the tangent conical surface.

Tangent cylindrical surface is tangent to the surface of revolution in the parallel circle, in the points of which there exist common tangent planes to both surfaces parallel to the axis *o* of the surface of revolution. Normals to the surface of revolution in the points of this parallel circle form the plane perpendicular to the axis of revolution.

Parallel circle is called the** neck **circle, if tangent planes to the surface of revolution in the points on this circle are parallel to the axis of revolution and the tangent cylindrical surface is located inside the surface of revolution.

Parallel circle is called the** equator **circle, if tangent planes to the surface of revolution in the points on this circle are parallel to the axis of revolution and the tangent cylindrical surface is located outside the surface of revolution.

Parallel circle is called the** crater **circle, if tangent plane to the surface of revolution in the points on this circle is perpendicular to the axis of revolution (circle is the set of parabolic points on the surface) and normal to the surface of revolution in the points of this parallel circle are parallel to the axis of revolution and form the normal cylindrical surface.

**2. Views of the surface of revolution**

Under the Monge method, we usually project surfaces of revolution in the position of the surface axis parallel to the coordinate axis *z*. Ground view of the surface of revolution appears as the part of the plane inbetween two concentric circles, which are views of the surface parallel circles of the extremal radii (minimal and maximal), Fig. 4. 46. All parallel circles have their ground views in the circles with the centres in one point, which is the ground view of the axis of revolution. Front views of these circles are line segments perpendicular to the view of the axis of revolution, parallel to the coordinate axis *x*_{1,2}. Front view of the surface of revolution has the outline in the view of the principle meridian located in the plane m // n, or also in the pair of line segments. These are front views of parallel circles located in the planes perpendicular to the axis of revolution and parallel to the ground image plane in the extremal distances. Principal meridian can be constructed as the set of points. Any parallel circle intersects principal meridian in two points, which are projected in the front view to the end points of the line segment - front view of the parallel circle. Front views of these intersection points are located on the front view of the parallel circle plane (parallel to the ground image plane), which is the line parallel to the axis *x*_{1,2}.

**3. Point on the surface of revolution and tangent plane to the surface in this point**

In any point *T* on the surface of revolution

there is located the parallel circle *k*

(intersection of the surface and the plane p* ^{T}* perpendicular to the axis of revolution

and the meridian

(intersection of the surface and the meridian plane m

Tangent plane t to the surface in the point

is determined by tangents

Ground view of the tangent

front view is a line parallel to the axis

Ground view of the tangent to the meridian

In the construction of its front view we will use the conical tangent with the vertex

which can be found as the intersection point with the tangent to the principle meridian in the point on the parallel circle

Point

Tangent

Tangent lines to all meridians on the surface of revolution in the points on the parallel circle

are passing through the point

Tangent

Normal to the surface, perpendicular to the tangent plane,

is projected in the ground view to the line perpendicular to the view of the tangent

Front view can be constructed from the normal conical surface.

Vertex

the right angle of the normal

Normal

**4. Intersection points of line and surface of revolution**

Frontal projecting plane c of the given line *a* perpendicular to the frontal image plane can be used for the construction. Intersection points of the line *a* and the curve, in which the plane c intersects the surface of revolution, are the intersection points of the line and the surface, *a* ÇF={*Q*, *R*}. Ground view of the intersection curve can be constructed as the set of points located in the plane c , on principle lines of the first frame. Front view is a line segment coinciding with the front view of the line *a*. Intersection points can be reached in the ground view. Visibility of the line with respect to the surface of revolution can be attached according to the visibility of the intersection curve (in the ground view) and with respect to the position of the intersection points and the principle meridian (in the front view), Fig. 4. 50.

**5.**** Planar intersection of the surface of revolution and tangent line to the intersection curve i****n one ****point**** **

Planar intersection of the surface of revolution by the plane a is the curve symmetric with respect to the line ^{1}*s*, which is the intersection line of the intersection plane and the plane of symmetry s of the surface perpendicular to the intersection plane.

Intersection points of the line ^{1}*s* and the surface are points on the intersection curve located in the minimal and maximal distance from the ground image plane p. They can be constructed as intersection points of the line ^{1}*s* and the meridian *m*^{s} ,

in which the plane s intersects the surface. Meridian *m*^{s} is congruent to the principle meridian *m*, which is projected in the front plane in the true size. Revolving the meridian *m*^{s} and line ^{1}*s* in the plane s to the plane m of the principle meridian the extremal points can be determined in the front views. All other points on the intersection curve can be determined as intersection points of parallel circles on the surface of revolution in the planes * ^{i}*l // p with the principle lines of the first frame

Fig. 4. 49