Some Interesting Curves

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Some Interesting Curves John D Barrow
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Swiss Re Building 30 St. Mary Axe The
Gherkin Norman Foster Partners
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The Swiss Re Building 180 metres high 40
floors 2003
Design Factors Sky visible Low winds on
ground Slow and smooth airflow Wedges bring in
air and light Six on each floor, Offset creates
spiral effect Helps bring air in All surfaces
flat cheaper!
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Torre Agbar Barcelona (2005) 142m
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Tatiana
Tatianas House
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The San Francisco Zoo Disaster
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y
Projectiles
v?
x
? u
Launch velocity (u, v) x ut y vt - ½ gt2
vx/u ½ gx2/u2 y vs x is a parabola dy/dt 0
at maximum height tmax v/g ymax ½ v2/g
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Mt Etna
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Crouching Tiger
h
V
x
V2 g (h ?(h2 x2))
Over short distances on the flat a tiger can
reach top speeds of more than 22 metres per sec
(ie 50 miles per hr). From a 5 metre start it
can easily reach a launch speed of 14 metres per
sec.
h 3.8 metres x 10 metres
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Only V gt 12 metres per sec launch speed needed
for the tiger to clear the wall
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The hanging chain, catenaria Leibniz and
Huygens 1691
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y Acosh(x/A) ½ Aex/A e-x/A
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The portion AP is in equilibrium under the
horizontal tension H at A, the tension F
directed along the tangent at P, and the weight
W of AP. If the weight of the string is w per
unit length and s is the arc AP, W ws and
from the force triangle, tan ? ws/H s/c,
where c H/w is called the parameter of the
catenary is determined by dy/dx s/c With
solutions    y c cosh(x/c)        s c
sinh(x/c)         
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Half A Catenary
The Rotunda was originally a tent put up in
London as part of the festivities to celebrate
the defeat of Napoleon. Designed by John Nash, It
was moved to Woolwich in 1816 and converted in
1920 into a permanent structure. It is now the
Royal Artillery Museum.
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Inverted Catenary Arches
The Gateway Arch, St Louis, MS 630 ft x 630 ft y
-127.7 ft cosh(x/127.7ft) 757.7ft Robert
Hooke 1671 Latin anagram (revealed in 1705) As
hangs the flexible chain, so inverted stand the
touching pieces of an arch.
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Can You Ride A Bike With Square Wheels?
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Stan Wagon Demonstrates
For the rolling square the shape of the road is
the catenary truncated at x sinh-1(1)
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For regular n-sided polygonal wheels the curve of
the road is made from catenaries with y
-Acosh(x/A) A Rcot(?/n)
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Clifton Suspension Bridge (1865)
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Suspension Bridges are Parabolas
Constant weight per unit length p
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Gustave Eiffels Tower (1899) moulded in a way
by the action of the wind itself
300m 894 ft high
f(x) ?x f(t) dt f ?(x) ?x (x - t) f(t) dt ?
f(x) Aebx
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Watkins Great Wembley Folly
Sir Edward Watkin, Chairman of Metropolitan
Railway saw Eiffels 894 ft Tower He wanted a
bigger one (1200 ft) on his land in Wembley
Park Eiffel refused. Benjamin Baker completed
stage 1 (155 ft) in Sept 1895. Opened 18th May
1896 but never went higher marshy ground and
shifting foundations. Tea shop for the new
Underground Station, few visitors Declared unsafe
in 1902. Demolished 1904-7. Iron sold for
scrap. Wembley Stadium built on the site in the
1920s
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Roller Coasters
Millennium Force, Cedar Point
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A Tale of Two Forces
You feel Force of Gravity Weight Mg ?
v
You feel radially outward Centrifugal Force Mv2/r
Mv2/r
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Staying in your Seat at the Top
Radius r
Fall from height h under gravity from rest ½mVb2
mgh At bottom Vb ?2gh
Ascend to the top of the circular loop of radius
r. Arriving there with speed Vt needs Energy
2mgr ½ mVt2 So mgh ½ mVb2 2mgr ½mVt2
Net Upward Force on rider (mass m) at top
mVt2/r mg gt 0 So we need Vt2 gt gr or you fall
out of the car ! h gt 2.5r
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Staying Conscious At the Bottom!
If h gt 2.5r you reach the bottom with speed Vb
?(2gh) gt ?(2g?2.5r) ?(5gr)
The net downward force on you at the bottom will
be Weight Centrifugal force mg mVb2 /r gt mg
5mg 6mg
A 6-g force will probably render you
unconscious Oxygen would not get to the brain
Circular roller coasters seem to fail their Risk
Analysis
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A Recipe for Success
We want Vt2 /r big at the top to hold us in But
Vb2 /r small at the bottom to reduce the g force
on the riders
Make r small at the top and big at the bottom
Ellipses first used in 1901 at Coney Island
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Clothoid loop
Shockwave roller coaster at Six Flags Over Texas,
Arlington Werner Stengels first use of the
Clothoid in 1975
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Clothoid loop
Loopen at Tusenfryd in Norway (Vekoma, Corkscrew,
1988)
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Clothoid curvature varies linearly with arc
length t
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Motorway Junctions
    An arc of a clothoid has variable
curvature, proportional to the distance along the
curve from the origin. It provides the
smoothest link between a straight line and a
circular curve. It is used in roads and
railroads design the centrifugal force actually
varies in proportion to the time, at a constant
rate, from zero value (along the straight line)
to the maximum value (along the curve) and then
back again to zero.
A vehicle following the curve at constant speed
will have a constant rate of angular
acceleration.
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At constant speed you can simply rotate the
steering wheel at a constant rotation rate. If
the bend was a different shape then you would
need to keep adjusting the rate of movement of
the steering wheel or the speed of the car
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Möbius and His Bands
August Möbius, notebook 1858
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Möbius Belts, Tape-drives and Conveyor belts
US Patent 3991631
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The Möbius Universal Recycling Symbol
Not a trademark!
Gary Anderson, Student at USC, design competition
winner, 1970
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Taiwanese Recycling Symbol
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Maurits Escher, woodcut Moebius Strip II (Red
Ants), 1963
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Robert Wilson, Fermilab, Batavia. Illinois
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