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Intersection of two superficial with generatrices curves:

We place the case (fig. 5) of two superficial ones with the aces to, b
incidents and pertaining to a profile plan. The quartica of
intersection is digrammica and symmetrical regarding two plans: one is
g, the other is b, passing for C and with perpendicular giacitura to
the axis b, of the Taurus.

For the determination of the quartica formed in this case from the
common points to the parallels of the Taurus and those of the surface
to vertical axis, we use of spheres secanti members of the women's
army auxiliary corps both the superficial ones. The said spheres have
the center in the point of encounter C between the aces, therefore
ripresenta the case of intersection between superficial with
coinciding aces (because the axis of the Taurus coincides with one of
the infinites aces of sfera).cioè the permission, deduce that the
intersections between the spheres members of the women's army
auxiliary corps and the superficial ones are circumferences. In the
case in examination the intersection between the spheres members of
the women's army auxiliary corps with the Taurus, is circumferences
that belong to the frontal plans and therefore has 2° the projection
in true shape. In 1°P.O. projections of the diameters are introduced
like segments ().

In the case of intersection between the said spheres with the surface
to vertical axis, the ciconferenze-sections belong to horizontal plans
and therefore they have the 1°P.O. in true shape, and the 2°P.O. like
segments.

The points of the quartica are therefore individualistic it is in
1°che in 2°P.O. like common points to the sections produced from every
single sphere member of the women's army auxiliary corps with the two
superficial ones.

As an example the point and, belongs is to the Q parallel of the
Taurus, than to the parallel and, sections, respective, of the sphere
with the Taurus and the main surface. The beam of this secante sphere
is segment C?D, since D belongs to the same relative Q
parallel-secante to the Taurus.

The location of the points, To, B, of principle and minimal quota the
quartica (referring only to the skillful branch) is immediate and it
does not demand the aid of secanti spheres, since sayings points
belong slowly to the frontal plan b (of longitudinal simmetria) and
therefore 2° the P.O. of the sections coincides with the second
projection of the contours allies by marriage of both the superficial
ones, respective, the parallels, D, F, of the maximum and minimal beam
of the Taurus and the H generatrix of the surface to vertical axis.

Intersection of two superficial ones, of which one to rectilinear
generatrix and the other to generatrix curve.

*
Between a cylinder and one sphere.
Between a cylinder and one ellissoide.

Intersection between a cylinder and one sphere

In several the cases the type of intersection curve depends is from
the mutual position of the aces of the two superficial ones that give
the measure of their diameter. The cases are following:

* It is placed that the cylinder is posizionato in such way
(fig.6), to have all the secanti generatrices and the axis for center
c of the sphere.

The quartica of intersection that is obtained is digrammica, composed
from two symmetrical coppers regarding the profile plan d and
therefore the following descriptions are reported to a single branch.
In the conditions of giacitura in examination he is convenient I use
it of the secanti plans is facades that horizontal, since these plans
sezionano the second cylinder and the second sphere circumferences.

Remarkable points of the quartica of intersection in the
rappresentazione in P.O.:

1. The points of F ceiling, and pertaining to the generatrices of
contour appearing of the cylinder in first P.O.
2. The points ceiling B, D pertaining to the generatrices of
contour appearing of the cylinder in second P.O.
3. The points of minim and principle quota the quartica that in
this case coincides with the B points, and, since the secante plan is
frontal.
4. The points G, H pertaining to the plan g of longitudinal
simmetria of the quartica, (you see 3°P.O.), and that they represent
with the symmetrical relati ones to you of according to branch
respective the nearer and farther points of the quartica regarding the
plan of cross-sectional simmetria .

Preliminary notes concluded necessaire are proceeded to the location
of the aforesaid remarkable points in the space and the orthogonal
projections.

1. In the 1°P.O the 2 points are attempted in which the quartica
the appearing contour of the cylinder is tangent. These two points
and, F are in common the circumference g and the two generatrices b,
dvengono determine to you by means of the horizontal plan to, that
seziona respective is the sphere is the cylinder.


1° the P.O. of the circumference, g1 has the individualistic
diameter in 2°P.O like segment delimited from the points of incidence
between t"to and the appearing contour of the sphere.

The 1°P.O. of the generatrices, b1, d1 coincides with 1° the
P.O. of the generatrices of contour appearing of the cylinder. 2° the
P.O. b2, d2 it coincides with the second trace of the plan to(slowly
projecting in 2° projection).
2. Others 2 remarkable points are those in which the quartica he is
tangent to the appearing contour of the cylinder in 2° orthogonal
projection. Sayings head are B, D, common to the circumference f and
to the generatrices h, i , it determines to you by means of the
horizontal plan b, that seziona respective the sphere and the cylinder.


The 1°P.O. of the circumference, f 1 coincides with t' b (slowly
projecting in first projection).

The 2°P.O. f 2, have individualistic diameter in 1°P.O like
segment delimited from the points of incidence between t"to and the
appearing contour of the sphere.

1° the P.O. of the generatrices, h1, i1 coincides with first
trace of the plan, t' b. 2° the P.O., h2, i2 coincides with 2° the
P.O. of the generatrices of contour appearing of the cylinder.
3. The G points, H of the quartica are respective the point more
close and that more far away, regarding the plan of cross-sectional
simmetria. Such points are in common all circumference n and to two
generatrices m, l. Both the sections are executed with the plan g
(slowly of cross-sectional simmetria), that it contains the axis of
the cylinder and passes for the center of the sphere.

The third projection of the said sections coincides with t' ''g. The
first and second P.O. of the circumference,n1and n2, would be
ellipses, therefore in order to find of the true shape is carried out
the ribaltamento of the secante plan g on one of the projection plans
or on a plan to they parallel; in this case ribaltando g on the
profile plan d the ribaltate sections are had, n * and l * , m *, from
whose intersection the G points are had *,H *. The distances G *-G3
and H *-H3 are equal to the distances elapsing between the 2°P.O. of
the points, G2,H2 and the second trace of the plan, t"d ,
accommodating said ribaltamento; you notice yourself that 1° and 2°
the P.O. of the G points,H they are equally individualistic assuming
of the horizontal secanti plans, whose traces are individualistic in
3°P.O., since pass for G3 and H3. With analogous procedure to that one
of point 1 the aforesaid points are characterized.

Development of one quartica of intersection between two resisted
circular cones


the technique illustrated here has the scope to explain in generic
way of as it can be developed one quartica of intersection between two
circular cones rectums, revolt mainly to persons has one good
acquaintance of the methods of rappresentazione of descriptive geometry.

the quartica of intersection between the two superficial auxiliary
conic sections has been found using of the plan passing for concerns
us of the two cones. The points of intersection between the several
complanari generatrices auxiliary goddesses every plan are those della
quartica try to you.

in this case the type of quartica (Q), is digrammica, that is composed
from two coppers, that we can simbologiare in this case, Qi that
inferior and Qs that advanced one.



Orthogonal projections "method of monge"
Procedure for the development of two coppers of the quartica Qi, Qs

- to divedere the circumference (dirittrice of the vertical cone) in
equal parts, as an example 10

- we consider that the perimeter of the circonfenza = H

- to design an arc of circumference To with equal beam to the
generatrix of the cone and has the equal perimeter to H.

- to divide the arc To in 10 parts (the same utlizzato number of parts
previously) and they join to you with the apex of the cone

to calculate the lengths of the segments of everyone of the 11
generatrices, is those reported to the advanced part of the quartica
Qs, is those of patre the inferior Qi. In order to calculate the
generatrices effetuano several the ribaltamenti in first slowly
vertical projection of ciscun conteneti the same generatrices.





development of the quartica is reported to the superficial ones of
the cone to vertical axis


axonometric sight
---

Intersection between superficial with rectilinear generatrices

1-3- Intersection between superficial with rectilinear generatrices:

1-3-1- Intersection between two cones (fig. 2).

We place the case of two resisted circular cones with aces (to, b)
incidents and parallels respective to the two main plans of projection
(p1,p2).

In order to find the common curve to the two superficial ones,
sezionano the two cones with a bundle of auxiliary plans passing for
vetrici V and and: everyone of these plans cuts the two cones, second
two generatrices. Made exception for the two plans that are tangent,
respective to one or to the dark cone. The common points to the
complanari generatrices constitute the quartica (digrammica) of tried
intersection. To such aim, the two join concern V second to us and and
straight r(the common one to all auxiliary plans), that they intersect
with the plans of the guiding ones, p1and to, in the T' points r, G.

Some of the remarkable points of the quartica of intersection are
wanted to be found, like the F points, H, in which the generatrices of
the cone of apex V they are tangent to the quartica. One assumes the
auxiliary plan g for r and tangent the cone of apex V, slowly
characterized it is from straight r that from l (the straight one
passing for the G point and tangent, in FE, the director d of the cone
to horizontal axis). Therefore the tangent generatrix d of the cone to
horizontal axis with the plan is constructed g, joining the apex and
with point FE of the horizontal director.

Aforesaid the straight one l intersects the plan of the director of
the cone of apex V in the T' point l , than combined with T' r it
determines the straight one of intersection t'g of the auxiliary plan
with that one of the director (p1). Once it characterizes the points
to you Fv 1 and H v 1 , incidence between t'g with the director QV1,
is combined with Vand cosi obtain two generatrices m, f, sections of
the auxiliary plan g with the cone to vertical axis. The common points
between generatrices m, f and that one of ceiling d are the points try
to you.

In order to complete the construction of the quartica, it is necessary
to assume other auxiliary plans, until determining a sufficient number
of points. In the figure in examination they are determines other
points to you like To, B (those of principle and minimal quota),
common in this case to the generatrices h, g and and, n, sections
respective of the two cones with the frontal plan b (containing the
aces of the two superficial ones).

Famous: since the quartica is symmetrical, regarding the plan b, has
referred to one single part for the determination of the aforesaid
points of the quartica.

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1-3-2- Intersection between two cylinders (fig.3).

In order to find the secanti quartica of intersection, auxiliary plans
to assume: they are those parallels to the aces of the two cylinders,
since contain you concern to us improper of the two cylinders.

1-3-3- Intersection between a cone and a cylinder (fig. 4)

The quartica of intersection in this case, is determined assuming of
the auxiliary plans passing for the apex of the two superficial ones,
is worth to say those plans that they have in common the straight one
r, that one passing for the apex of the cone and parallels to the axis
of the cylinder.

Famous: the cylinder is considered a cone with the improper apex and
therefore in everyone of the three previous cases, the secanti plans
are those passing for concern us of the two surface considered.

It continues............ Successive page Þ