Rigging Math
(Made
Simple)
A
Primer by
Delbert L.
Hall, Ph.D.
ETCP Certified
Rigger
ETCP
Recognized Trainer
Lesson 13: Three-Point Bridle Lengths
Bridles are very common in many types of theatrical rigging, especially
arena rigging, and are used to create hanging points in locations where
there are no attachment points directly above from which to hang.
The simplest, and most common type of bridle, is a two-point bridle.
A two-point bridle uses two cables
(or legs) to create a new hanging point where the two legs join.
This junction point is on the same plane as the attachment points for
the two legs, just lower and somewhere between the two existing
attachment points. The lengths of the two legs of the bridle
determine both the height and the vertical position of the bridle point
(junction). The Pythagorean Theorem is commonly used to calculate
the lengths of the two bridle legs.
A three-point bridle is used when
the desired bridle point is not directly between two existing
attachment points and must be positioned between three points.
This is common in spaces where the existing attachment points are
scattered about. When the three legs are the proper length, a new
hanging point is created that is in the desired location above the
stage.
Calculating Three-point bridle lengths
Bridles are very common in many types of theatrical rigging, especially arena rigging, and are used to create hanging points in locations where there are no attachment points directly above from which to hang. The simplest, and most common type of bridle, is a two-point bridle.
The math for calculating the bridle lengths on a three-point bridle is
a little more complicated than calculating the bridle lengths for two
point bridles. With two point bridles, you have only three points
with which to be concerned: the attachment point for each bridle leg
and the bridle point where the two legs meet. Since all three of
these points are on a single plane, you only need to know the
horizontal and vertical distances of the hanging points from the bridle
point in order calculate the two bridle lengths. Therefore, you only
need to know four numbers - really simple.
A three-point bridle problem is
three-dimensional. You must know the X, Y and Z coordinates of
the attachment points for the three bridle legs and the bridle point
(12 coordinates numbers). See why it is more complicated?
The first trick to solving this problem is to
collect and organize the data. The best way is to complete a coordinate
table with the needed data, such as the one below:
P1 X: ___________ Y: ___________ Z: __________
P2 X: ___________ Y: ___________ Z: __________
P3 X: ___________ Y: ___________ Z: __________
P4 X: ___________ Y: ___________ Z: __________
Note: P1, P2, and P3 will be the hanging points for our three bridle
legs (L1, L2 and L3) and P4 will be the bridle point (where the three
legs meet).
Look at the hanging plot (plan
view) below. In this plot you see that the three attachment
points are located on two beams and the bridle point is between
then. Each attachment point on the plot, as well as the
bridle point, is marked with its X and Y coordinates, in
parentheses. I called the lower left hanging point on my plot 0,0
so that all of the other coordinate points are positive numbers (since
they are in Quadrant I) and relative to this point. You can set
up your coordinate system whatever way makes the most sense for you.
P1 X1: __0____ Y1: __0______ Z1: _________
P2 X2: __0____ Y2: __12_____ Z2: __________
P3 X3: _16____ Y3: ___6_____ Z3: __________
P4 X4: __8____ Y4: ___7_____ Z4: __________
Next, we need to input the Z coordinates. These are the heights
of the points. So, if the bottoms of the two beams are 50 feet
above the deck and the bridle point is 35 feet above the deck, the
completed coordinate table would look like this:
P1 X1: __0____ Y1: __0______ Z1: ___50____
P2 X2: __0____ Y2: __12_____ Z2: ___50____
P3 X3: _16____ Y3: ___6_____ Z3: ___50____
P4 X4: __8____ Y4: ___7_____ Z4: ___35____
Now that the table is complete, it is time to do some math. The
formulas for computing the lengths of the three bridle legs (L1, L2 and
L3) are:
This is really just a variation on
the Pythagorean Theorem. The big difference is that you must
subtract the appropriate axis coordinate for the bridle point (P4) from
the same axis coordinate for the three attachment-point coordinates
before you square it. If you are taking the ETCP exam to become a
certified rigger, these formulas are listed on the Formula Table that
you will be given (just be sure you are able to recognize them from the
many other formulas on the sheet). After you work a few problems,
this is actually a pretty easy equation to remember.
L1 = 18.38 feet
Another Method
There is another way to calculate three-point bridle
lengths, one that uses the Pythagorean Theorem Š the same formula used
to calculate the lengths of two-point bridles. The
trick to using this simpler formula is that you must use it twice to
calculate the length of each bridle leg. LetÕs look at how this
method works.
Using the coordinates that we used above, letÕs look
at an elevation view of Leg 1, and what we know and what we do not know
about Leg 1.
To compute the length of the bridle leg (the distance between P1 and
P4) we need to know both the vertical and horizontal distances between
these two points. We know that the vertical distance is 15 feet
(50 feet minus 35 feet), but we need to figure out the vertical
distance before we can use the Pythagorean Theorem to calculate the
hypotenuse of this triangle (the bridle length).
Fortunately, there is an easy way to find this distance.
The plan view below is the same as one used earlier,
except I have overlaid it with three right-triangles (one Green, one
Red, and one Blue). The hypotenuse of each of these triangles
correlates with one of the three bridle legs and the two vertices are
on the X and Y axis.
Note: Creating these three right triangles (where the hypotenuses
correlates with the bridle legs) is a very important step. Look
carefully at the drawing below and understand how these triangles were
created. Without these triangles, and knowing the lengths of
their vertices, you cannot calculate the horizontal lengths.
Look at the Green triangle. The hypotenuse of this triangle (the
dashed line between P1 and P4) correlates with the horizontal line in
the preceding drawing. As you see, the lengths of the two
vertices of this triangle are 8 feet and 7 feet. By using the
Pythagorean Theorem, we can calculate the horizontal distance between
P1 and P4 (the hypotenuse of this triangle).
feet
Now that we know the horizontal distance, use this
number with the vertical distance (15 feet) to determine the length of
the bridle L1.
feet
This is exactly the same length for L1 that we
calculated using the first method. As you see, by using this
simple formula (twice) we can calculate the length of a three-point
bridle.
By using 5 feet and 8 feet for the verices of
the Red triangle, the horizontal distance for L2 can be calculated as
9.43 feet. And by using 8 feet and 1 foot for the vertices
of the Blue triangle, the horizontal distance for L3 can be calculated
as 8.06 feet. These numbers, along with the vertical height of 15
feet, can be used to calculate the lengths of L2 and L3.
