Embed Size (px)
Citation preview
SY
MM
ET
RY
AX
IS
VASE MADE WITH TANGENCY BETWEEN LINES AND CIRCLES
Look at how these lines and curves have been connected. Remember that the sign meansthat there are two perpendicular lines (in this exercise, if this symbol doesn’t appear it means thatthe lines are not perpendicular). We will start drawing this part on our drawing paper, using thecompass so as to translate the enlarged dimensions.
THIS DRAWING IS MADE ON A 3:4 SCALE. WE HAVE TO PUT THESE DIMENSIONS ON A4:3 SCALE SO AS TO RESTORE THE OBJECT’S ORIGINAL DIMENSIONS, STARTING WITH THE AXIS AND KEEPING IT CENTRED. TO FIND THE TANGENCY POINTS, WE HAVE TOTRANSLATE THE ANGLES THAT THE RADIUSES MAKE BETWEEN THEM.
S=3:4
E=3:4
INITIAL COPY (PRINT IT)
SY
MM
ET
RY
AX
IS
3 c
m 4 c
m
V
To apply to this drawing a 4:3 scale, we use the following construction,
based on the Thales theorem:
On a new sheet, we draw a long horizontal line, and from it, we draw two
perpendicular segments, one of them 3cm length and the other one 4 cm
length. These segments have to be parallel and close enough to produce a
line that doesn’t go outside the sheet when joining their ends.
SEPARATE SHEET
4:3 PROPORTION
A
B
3 c
m 4 c
m
V
4:3
LIN
ECompass
We join the vertex V with the mark made with the compass on the vertical line which goes
from the 3 cm mark, and we lengthen it to the vertical line drawn in the right side, where we
get the dimension already enlarged to a 4:3 scale, which is the one we have to use in our
drawing paper.
(EN
LA
RG
ED
DIM
EN
SIO
NS
)1.
2.
HOW TO ENLARGE THE DIMENSIONS TAKEN FROM OUR INITIAL COPY
SHEET SO AS TO PUT THEM ON OUR DRAWING PAPER.
On this line (the one with the mark of 3 cm.) we take the dimensions from
the initial copy, from the foot of the perpendicular.
Foot of the perpendicular (where two perpendicular lines cross)
SEPARATE SHEET
On this
line, w
e b
ring the d
imensi
ons
from
the in
itial c
opy
VASE, TANGENCY
WE DRAW THE MARGIN AND THE BOX
VASE, TANGENCY
VASE, TANGENCY
ENLACES Y TANGENCIAS
VASE, TANGENCY
AX
IS O
F S
YM
ME
TR
Y
VASE, TANGENCY
E=3:4
On the initial copy, we take with the compass the distance between the two foots of the
perpendicular lines to the axis, the upper one and the lower one. Then we have the altitude of
the figure.
INITIAL COPY
3 c
m 4 c
m
V
SEPARATE SHEET
4:3
EN
LA
RG
ED
DIM
EN
SIO
NS
We transport that dimension to this line
OR
IGIN
AL D
IME
NS
ION
S (
FR
OM
TH
E IN
ITIA
L S
HE
ET
)
3 c
m 4 c
m
V
SEPARATE SHEET
And we enlarge it
EN
LA
RG
ED
DIM
EN
SIO
NS
OR
IGIN
AL D
IME
NS
ION
S (
FR
OM
TH
E IN
ITIA
L S
HE
ET
)
V
Taking more than a half of the length, we trace two arcs on the left to
draw the perpendicular bisector of this enlarged segment.
SEPARATE SHEET
SEPARATE SHEET
V
V
SEPARATE SHEET
V
SEPARATE SHEET
V
SEPARATE SHEET
V
We take one half of the segment.
SEPARATE SHEET
And we translate it up and down on the axis, from its center.
VASE, TANGENCY
VASE, TANGENCY
VASE, TANGENCY
VASE, TANGENCY
VASE, TANGENCY
INITIAL COPY
AX
IS O
F S
YM
ME
TR
Y
E=3:4
With the compass, we take the length which goes from the axis to the first
tangency point, on the upper horizontal line in the copy sheet.
And once enlarged in our separate sheet, we translate it to its
corresponding line in our drawing paper (always with the compass).
VASE, TANGENCY
VASE, TANGENCY
LINE THE SET SQUARE UP WITH THE HORIZONTAL
We trace a perpendicular line from the marked point.
VASE, TANGENCY
On this perpendicular line and from its foot, we translate the length of the
first circle radius from our copy, the smallest one, after having enlarged it
in our separate sheet. We also lengthen the perpendicular line up.
VASE, TANGENCY
We draw the first circle, which has to be tangent to the upper horizontal
line.
VASE, TANGENCY
E=3:4
INITIAL COPY
In our copy sheet, we lengthen the vertical
radius of the small circle upwards, just to
be able to work better, as we have to copy
the angle that the two blue radiuses form
in our drawing paper.
E=3:4
1
2
In order to copy this angle, we draw first an
arc that crosses the two enlarged radiuses,
with the most comfortable compass opening
and taking center in its vertex. We have to
focus on the points 1 and 2.
INITIAL COPY
1’
With the same compass opening, we trace an equal arc in our drawing
paper, taking center on the vertex of which will be our copied angle, and
we mark the cutting point with the enlarged vertical radius (point 1’).
VASE, TANGENCY
E=3:4
1
2
Then, with the compass, we take the
distance between the points A and B
in our copy sheet.
INITIAL COPY
1’
And we take this distance (chord) from the point 1’ drawing an arc which
crosses the previously drawn arc, getting the point 2’.
2’
VASE, TANGENCY
1’
2’
Drawing a right line that joins the vertex of the angle with the
point 2’, we have already copied the angle.
VASE, TANGENCY
E=3:4
In our copy sheet, we take the length of the
radius on the next circle with the compass.
T
And, after having enlarged it in our separate sheet, we transport it on the
side of the copied angle from the point T, which will be the tangency one
between the two circles.
VASE, TANGENCY
T
Taking center on the previous mark and with tthe same radius, we draw a circle which will
make contact with the previous one in the point T.
VASE, TANGENCY
E=3:4
Then, we have to copy the angle
formed by the two blue radius so
as to keep drawing.
E=3:4
1
2
We will remind you how to do
it: taking center on the vertex
of the angle to be copied, we
trace an arc and we focus on
the points in which it cuts the
two sides, points 1 and 2. The
compass opening Will be the
most convenient for us.
1’
Without changing the chosen compass opening, we trace an equal arc in
our drawing paper, and we we focus on the point in which this arc cuts the
only side we have for the time being.
VASE, TANGENCY
E=3:4
1
2
In our initial copy, we take the
distance between the points 1
and 2 with the compass.
1’
With this compass opening, we take center on the point 1’ from our
drawing paper, and we trace an arc that cuts the first one. This cut-off
point is 2’.
2’
VASE, TANGENCY
1’
Joining the center of the initial arc with the point 2’, we have already
copied the angle. We lengthen this second side of the angle which passes
through the point 2’ to the cut-off point with the circle.
2’
VASE, TANGENCY
LINE UP WITH THIS RADIUS
Then, we trace the tangent line to this circle at
the cut-off point between it and the lower
radius. As we know that the tangent line has
to be perpendicular to the radius, we line the
set square up with it.
VASE, TANGENCY
We slide the 45° set square not letting the 60° set square move until we
see the cut-off point (foot of the perpendicular). Then, we hold the 45° set
square and we draw with wit the other leg, the one we didn’t line up
before, the line perpendicular to the radius, which is tangent to the circle
at the chosen point.
VASE, TANGENCY
On the drawn tangent line we transport the legth between the two tangency points of this line
with the two circles. To do so, we take this length from our copy sheet and enlarge it in our
separate sheet first.
VASE, TANGENCY
From the mark we have just done, we draw a line perpendicular to the
previous one. Remember to do it using the two set squares.
VASE, TANGENCY
On the perpendicular, we transport the legth of the next circle radius, taken from our initial
copy and enlarged in our separate sheet
VASE, TANGENCY
With the same compass opening, we take center on the recently made mark, and we trace the
circle, which has to be tangent to the line.
VASE, TANGENCY
Using the same method than in the previous cases, we copy the angle that the two radius in
blue form. We take it from our initial copy sheet and we transport it to our separate sheet. We
enlarge the second side of the obtained angle, out of the circle.
VASE, TANGENCY
From the new tangency point, T, we transport with the compass the length of the following
circle radius, taken from our initial copy sheet and enlarged in our separate sheet.
T
VASE, TANGENCY
Taking center on the last obtained mark and without changing the compass opening, we draw
the circle, which Will be tangent to the previous one at the point T.
T
VASE, TANGENCY
We draw the following radius, copying the angle between it and the one that we already have.
We know how to do it.
VASE, TANGENCY
By the end of the radius we draw a perpendicular
line, which will be tangent to the circle…
VASE, TANGENCY
And we transport on it (as we did before, page 48)
the distance between the two tangency points,
taken from our initial copy sheet and enlarged in
our separate sheet.
VASE, TANGENCY
By the recent mark, we draw another
perpendicular line down, and from the vertex of
the right angle (foot of the perpendicular), we
transport the length of the last circle radius, once
taken from our initial copy sheet and enlarged in
our separate sheet.
VASE, TANGENCY
We draw one last circle, and trace a
vertical radius down (parallel to the
symmetry axis). From the end of that
vertical radius, we draw a line tangent
to the circle. This line will be horizontal,
which means perpendicular to the axis,
and we lengthen it until it gets this axis.
VASE, TANGENCY
E=3:4
FROM THIS PAGE, THE DRAWING IS MADE AT A SMALLER SCALE AND
WITHOUT A MARGIN, BUT THIS MUSTN’T AFFECT TO OUR EXERCISE IN
THE DRAWING PAPER, THAT STILL HAS TO BE DONE WITH THE ENLARGED
DIMENSIONS. THIS MEANS THAT FROM NOW, THIS IMAGE PORTRAYS OUR
DRAWING PAPER
We trace perpendicular lines to the symmetry axis from every circle
centers, and we lengthen them to the other side of the it.
E=3:4
A A’
B
B’
We find the symmetrical points of the circle centers, poking with thecompass into the cut-off point between the perpendicular line to the axisand this last one, taking the distance from this point to the original oneand transporting it to the opposite side of the axis, always on the perpendicularline to this one previously drawn.
E=3:4
A A’
B
B’
C C’
DD’
EE’
YOU DON’T HAVE TO DRAW THA WHOLE ARCS
E=3:4
A A’
B
B’
C C’
DD’
EE’
We draw the symmetrical circles to the ones with centers B and C, with the sameradius than the original ones.
E=3:4
A A’
B
B’
C C’
DD’
EE’
B’ radius +C’ radius
We are going to trace an interior tangent line to the circles with centres B’
and C’. To do so, we draw a radius on the circle with centre C’, and we
enlarge it to add the length of the radius of the circle with centre B’ next.
Once added the radius, we draw the circle with centre C’ and radius the
addition of the two said.
This is how we add
two radiuses
E=3:4
A A’
B
B’
C
C’
DD’
EE’
We join B’ and C’ drawing a segment.
E=3:4
A A’
B
B’
C
C’
DD’
EE’
Perpendicular bisector of B’C’
M
Tracing the perpendicular bisector of the segment B’C’, we can have its midpoint M.
E=3:4
A A’
B
B’
C
C’
DD’
EE’
M
We have to draw the circle with centre M and diameter B’C’, and mark the point N,where this circle cuts the previous one (the one with centre C’ and radius RB’+RC’).
N
E=3:4
A A’
B
B’
C
C’
DD’
EE’
M
N
We trace the line that joins the points B’ and N, which is tangent to the bigger circleat the point N.
E=3:4
A A’
B
B’
C
C’
DD’
EE’
NP
Q
From the points B’ and N, we trace two lines, the one from N going to the centreC’ until it cuts the next circle in the point P, and the one from B’ being parallel to NP,and ending at the point Q when cutting the circle.
E=3:4
A A’
B
B’
C
C’
DD’
EE’
NP
Q
We draw a line from the point Q to the point P. This line has to be parallel to theone defined by the points B’ and N, and it also has to be tangent to the two cirles atthe points Q and P.
A
B
B’
C C’
DD’
EE’
E=3:4
A A’
B
B’
C
DD’
EE’
T
So as to complete the symmetrical part of the figure, we draw the symmetrical circlesto the original ones. For the circle drawn in blue, we first join the centres C’ and D’through a segment which cuts the circle with centre C’ at the point T of tangency.Taking centre in D’ with the compass, we open it to the point T to draw the circle in blue.We make the same to draw the circle with centre A’. From A’ and from E’ we draw verticalradiuses, which have to cut the upper and lower horizontal lines at the tangency pointsTA’ and TE’.
TE’
TA’
E=3:4
A
B
B’
C C’
DD’
E=3:4
A A’
B
B’
C
DD’
E E’
We repeat the interior tangent lines tracing procedure with the circles
with centres D’ and E’.
E=3:4
A
B
B’
C C’
DD’
E=3:4
A A’
B
B’
C
DD’
E E’
We repeat the interior tangent lines tracing procedure with the circles
with centres D’ and E’.
E=3:4
A
B
B’
C C’
DD’
E=3:4
A A’
B
B’
C
DD’
E E’
We repeat the interior tangent lines tracing procedure with the circles
with centres D’ and E’.
E=3:4
A
B
B’
C C’
DD’
E=3:4
A A’
B
B’
C
DD’
E E’
We repeat the interior tangent lines tracing procedure with the circles
with centres D’ and E’.
E=3:4
A
B
B’
C C’
DD’
E=3:4
A A’
B
B’
C
DD’
E E’
We repeat the interior tangent lines tracing procedure with the circles
with centres D’ and E’.
E=3:4
A
B
B’
C C’
DD’
E=3:4
A A’
B
B’
C
DD’
E E’
We repeat the interior tangent lines tracing procedure with the circles
with centres D’ and E’.
E=3:4
A
B
B’
C C’
DD’
E=3:4
A A’
B
B’
C
DD’
E E’
We repeat the interior tangent lines tracing procedure with the circles
with centres D’ and E’. Although we have erased some lines here, we
won’t erase anything until having finished the drawing and made a
photocopy of it.
E=3:4
Once the drawing completed, we make a photocopy of it so as to be able to see the auxiliarypaths we have made. In our drawing paper, we use our technical pen (0,8mm) to draw in inkthe linked lines which form the outline of our vase. Once the ink is dry, we have to erase the remainig pencil and we can paint the figure the way we like.
