S u mmar y S h e e t - Bishop O'Dowd High School · 2018-02-27 · Team Control Number: 7423 Page 1...

Team Control Number: 7423

2017

20th Annual High School Mathematical Contest in Modeling (HiMCM) Summary Sheet


Problem Chosen: A

Summary Sheet

The Shooting StarTM drones developed by Intel® allowed cluster of drones to fly simultaneously

to create 3D-shape effects. These drones can be used to create fantastic light shows. While

demonstrating a combination of art and drone innovation, the aerial light show creates a new

experience for everyone. For our light show, there will be 171 light drones flying systematically

into the air to form special images, including a ferris wheel of happiness, dragon of courage, and

Golden Gate Bridge of hope.

Our mathematical model in “Geogebra”, in essence, is a beautiful animation accomplished by

utilizing parameterized functions. We will be using functions with parameters to define a point’s

coordinate. Let’s set point A’s coordinate to (sin(a), cos(a)). The sum of squares of the x

coordinate and y coordinate of A is equal to 1, thus the distance from A to the origin is always 1,

the fixed constant. When parameter a changes, point A would be moving on the path of a circle

with center(0,0) and radius 1.

With this generic model and other testings , we further explored the “fixed moving system” and

the “relative moving system”. A “fixes moving system” refers to moving as a whole, such as two

points moving together. A “relative moving system” refers to one moving on the basis of the

other, such as a point moves around a line . Using one generic model and two advanced models,

we eventually build up our fabulous Ferris Wheel, Dragon, and Golden Gate Bridge. In the

animation, The Ferris Wheel will be rotating by its own. The Dragon will be opening its mouths,

shaking its ears and backs. The Golden Gate Bridge will be expanding to twice its dimensions

and returning back to its original size. Our inspirational light show will give every audience an

unforgettable experience.


Table of Contents

Cover page and summary …………………………………………………………….…………. 1

Memo to Mayor …………………………………………………………….……….………….. 3

1-Introduction …………………………………………………………….……….……….……. 5

1.1 Background …………………………………………………………….……….….…5

1.2 Restatement of Problem ……………………………………………………………....5

1.3 Assumptions and justification ………………………………………………………...5

2. Flying System Description …………………………………………………………….….……7

2.1 Number of Drones …………………………………………………………….….….. 7

2.2 Place Setting of the Performance ….………………………………………………….7

2.3 Switching from Initial Plane to Display Plane ………………………………………10

3. Model of Displaying and Animation …………………………………………………………11

3.1 Generating Models …………………………………………………………….….…11

3.2 Time Parameter …………………………………………………………….………..15

3.3 Animation Video …………………………………………………………….……... 17

3.4 Ferris Wheel …………………………………………………………….…………...17

3.5 Dragon ……………………………………………………………..….….….….…...19

3.6 Golden Gate Bridge ……………………………………………………………..…..21

3.7 Combining time and Coordinate Parameters …………………………….…………21

3.8 Transformation ……………………………………………………………..….….…22

3.9 Transformation pathway as a function of time …………………………….….….…24

3.10 Calculating Velocity during animations and transformation ……………………....25

4. Requirements Specifications ……………………………………………………………..…..25

4.1 Duration of the Show ……………………………………………………………..…25

4.2 Safety Considerations ……………………………………………………………….27

5. Final statement ……………………………………………………………..….….….….…....27

6. References ……………………………………………………………..….….….….….….….28


Memorandum

To: Mr. Mayor

From: HiMCM 2017 Team 7423

Date: November 19, 2017

Re: Memo-Drone Light Show

We would like to share with you the results of our investigation on the possibility of adding an

outdoor aerial light show to our city’s annual festival. In order to give advice on whether or not

to do the aerial light show, we developed a model concerning several facts about the operation of

the drones that will allow this air show to happen in our city.

First of all, we considered the number of drones required to display and animate the images,

which includes a ferris wheel, a dragon, and the Golden Gate Bridge. We generated a model to

describe the movements and flight paths of the drones when launching and landing, and more

importantly, how to form different animations during the light show. We mathematically

described these aspects to ensure the possibility to create a program that enables the drones to be

safely operated, avoiding collisions. In addition to describing how the drones can be actually

controlled to perform an amazing light show, we also considered and evaluated the other

requirements for this 3-display light show, mainly, the conditions needed. The conditions

required for this air show is concluded by us proving that we are able to hold this aerial light

show during the festival.

For modelling the drones to form a ferris wheel, a dragon, and the Golden Bridge, we chose a

number of 171 drones. Their flight paths are described by using building a mathematical model

on Geogebra. We used several parameterized functions to animate our three objects on

Geogebra. The general rule for this model using Geogebra is easy to understand, by setting up a

parametrized function, the distance from a point will always stay the same, thus a change in

parameter will change the path that point moves. In the animation, the ferris wheel will be

rotating, the dragon will be opening its mouth, shaking its ears and backs, and the Golden Gate

Bridge will be expanding its dimensions larger. In addition to the flight paths and animation, we


also developed ways to avoid collision while operating the drones. All these ways based on our

Geogebra model gives us the ability to hold this light show in our city.

Another important aspect in our investigation is the safety considerations and other requirements

for this aerial light show. This includes the limitations in the space needed, the duration of this

show, and other considerations regarding the operation of the drones. To safely launch 171

drones, a square-shaped open area in needed. In order to ensure safe distance between each

drone, an area of 2400m2 is needed. This area can be at any open places, the Union Square in our

city will be the best choice in this case. The flight limitation is usually limited to 400 feet(129

meter) in U.S. territory, otherwise it will affect the actual airplane routes. Based on the battery

life of the drones, the light shows is generally to be expected to be less than 20 minutes, for the

animations we designed for the three object, the duration of the show is planned to be 3.5

minutes in maximum including the time for launching and landing.

We believe holding this aerial light show in our city will be an inspiring moment that shows a

perfect combination of art and technology. In reality, this newly developed light show performed

by hundreds of drones is a precedent, and practically it can be used to put ads in the sky, or be an

stunning performance. Compares to other public performances such as fireworks or parades, an

fabulous aerial light show is more meaningful by showing technology innovations.

The first public performance of drones light show took place last summer in Sydney during the

2016 Vivid Sydney event in Australia, accompanied live by the Sydney Youth Orchestra. It’s a

breathtaking experience for everyone witnessing the show, and this year, on the Coachella Music

Festival, a aerial light show performed by three hundred drones surprised the audiences. Their

posts on social media caught attentions from all over the world, causing people to be aware of

this revolutionary improvements on drones. Therefore, we recommend our city to host this aerial

light show considering the benefits to the audiences, and even to our city’s reputation.

Sincerely, Research Team 7423


1-Introduction

1.1 Background

Shooting StarTM drones has been developed by Intel® for aerial light shows. By using cluster of

these drones, fantastic light shows can be performed with only one pilot and a single laptop. In

2016, 500 of these drones performed a beautiful light show in the sky with only one person

controlling all of them. The fleet formed 3D shapes, spelled out recognizable words and painted

the number 500 in the air. This technology-powered light show demonstrates how art and drone

innovation can combine to create new enjoyable entertainment. The Intel team first set a

Guinness World Records title in late 2015 with a 100 drone light show outside of Hamburg,

Germany. Later the Drone 100 performed publicly during the summer 2016 Vivid Sydney event

in Australia. The optimized system used by the Intel Shooting Star drones ensures the show to be

a new experience for everyone.

1.2 Restatement of Problem

Our city started planning for the annual festival, and an outdoor aerial light show is considered as

a possible add-on to the festival. But to be able to perform this aerial light show, hundreds of

drones with LED light have to be controlled in a way that ensures safety while giving us an

amazing view. The Mayor asked us to investigate of this idea of using drones to create possible

sky displays, specifically, a ferris wheel, a dragon, and an image of the Golden State Bridge of

our city. Our objective is to create a plan showing that this aerial light show is possible to be held

in aspects including number of drones, initial location for each drone, the flight paths of each

drone, and how the animation is formed. Additionally, basic explanations about the conditions of

this aerial light show is also required to support our conclusion.

1.3 Assumptions and justification:

Assumption 1: Every drone used do not malfunction during the show. No spontaneous

shut-down. It will follow the designed pathway exactly.


Justification 1: Our job is to plan a light show. Therefore, functioning drones are fundamental

for the show to proceed.

Assumption 2: The wind speed during the performance will not go over the maximum wind

speed that a drone can bear. The weather is good during the performance and therefore the aerial

light show will not be affected.

Justification 2: Weather conditions should be normal so that the audience has a clear view of the

light show light. Days of unfavorable weather should be avoided from performance scheduling.

Assumption 3: The drone moves at a maximum speed of 3 meters per second.

Justification 3: According to the Intel(R) Shooting StarTM drone, 3m/s is the maximum speed for

the aerial light show drone.

Assumption 4: Space or landscape is not a limitation for the light show except for the legally

limited height.

Justification 4: Aerial light show is often performed in large and spacious area and therefore

there is always enough space.

Assumption 5: Drones that are being used are identical.

Justification 5: If the drones are not the same, it will be really hard to plan a light show.

Assumption 6: When transforming from one display to another, number of intersecting paths

generated using our model is smaller than 50.

Justification 6: Using the two way table from section 3.8, we are able to designate drones new

locations near their current ones, which efficiently decrease the number of intersecting paths. By

testing out smaller models, the maximum ratio of intersecting paths to total drone number is 1:3,

from which we base our assumption on.


2. Flying System Description

Display 1: Ferris Wheel

Display 2: Dragons

Display 3: Golden Gate Bridge

2.1 Number of Drones:

171 drones in total are needed to perform the aerial light show during the festival in the city. For

the first and third part of this light show, the Ferris Wheel and the Golden Gate Bridge, 3 drones

will be a group as an equilateral triangle with a side length of 0.5 meter. For the dragon

animation, each group will be 2 drones which are 0.5 meter apart from each other.

2.2 Place Setting of the Performance:


DP: Displaying Plane of the drones (x-y plane facing the auditorium)

IP: Initial Plane, initial location of the drones before launching (x-y plane parallel the ground)

FOV: Field of view of the audience

t1: The time needed for all of the drones to reach the display area to display the first image from

the ground

Planes building and dividing:

To perform a clear image of the displays for the audience, the displaying surface of drones

should be tilted. In this performance, the surface is 30 degrees tilted and should be perpendicular

to the audience’s eyesight. DP and IP are two separated x-y planes. All of the animations and

transformations between displays will be using DP as an x-y plane facing the auditorium. From

the audience’s view, the display will be mostly 2D, except for the transformation between

images which will be discussed in later section.

Display Height and Width:

The highest point of the display will be at settled at 120 meters from the ground because the

maximum height that is legal for drones to reach is about 121.92 meters. The width of the DP is

going to be 40 meters. One third of the height is a proper size for the displays. The width of the

displays will vary during the animations. The height of the lowest point of DP will be.

20 0 in( ) 5.3591 − 4 × s 3π ≈ 8 meters,

which is the distance between IP and DP. The distance between any of the two drones will not

be less than 0.5 meters to prevent them from being affected by each other and form causing

accidents.

Distance to the Audience

Since an image that is perpendicular to the audience will be clear to them and the display angle

tilted is 30 degrees, auditorium should be about

meters from IP.5.359 an(π/3) 47.8468 × t ≈ 1


IP Plane (Initial Location) and Space required:

FW: The display of the ferris wheel

Each dot (including both black and blue dots) represents a group of 3 drones in Figure 3.2_3 and

Figure 3.2_4 shown below. We are placing each group of 3 drones at where they will be

projected on the ground. By doing this, we can optimize t1, because they are starting from the

closest place and going through the shortest path way. Since the display is tilted for 30 degrees,

the width of IP (the distance from group J to group V in Figure, 3.2_4 below) is equal to half of

the width of DP (FW in this case). The initial placement of drones has a width 20 meters and a

length 40 meters. 800 square meters are needed for this initial setting.

P W os(π/3)I width = F width × c

(meters)P F W 0I width = 21

width = 2

Figure 3.2_3 is the drone placement we designed for the first display. In this case the width of

the display is equal to the length of the display. For Figure 3.2_4, x-axe is increasing toward the

left, y-axe is increasing toward the audience.


The light show requires the minimum total air space of about m3 from0 20 20 760004 × 1 × 1 = 5

ground to the highest point of the performance.

Considering safety issues, the actual air space needs to be bigger than the minimum size.

Therefore, we decide to make the size

m3.0 30 30 450005 × 1 × 1 = 8

2.3 Switching from Initial Plane to Display Plane:

DGi: the distance a drone group needed to raise to the required display position

Time Needed:

All units of 3 drones are launching at the same time. They are going straight up until they

reach their final position in the air. Therefore, t1 will be the time needed for the groups of

drones that are going to be the highest groups to reach their position. The distance is


equal to the height of the highest point of FW, which is 120 meters. Since the maximum

speed of drones are 3 meters per second,

t1 40≥ seconds.

40 seconds are needed for drones to reach their required position of Display 1.

Distance of Raising:

Using the the coordinates of the Ferris Wheel in the Appendix, the group of drones

Gi(x,y) need to raise:

.82(y − 2))tan(π/3)DGi≈ 1 − ( 1

3. Model of Displaying and Animation

“Geogebra” is an interactive graphing application which can picture the 2-dimensional and

3-dimensional objects on the basis of mathematical expressions. Our team used a combination of

the static points and the non-static points to build up the general display and animation of the

three final shapes, the Ferris Wheel, the Dragon, and the Golden Gate Bridge.

3.1 Generating Models

Single Point On a Circle Path

We divide 171 drones into 57 drone units, with 3 drones per unit. Every drone unit is a point,

which can either be with parameter, or without parameter. The points on the main structure of the

object is more likely to be parameterized, since it requires animation motion. The parameterized

function is crucial in the modelling.

To start with, we should first define our parameter “a” along with its boundary, our original

parameterized function is:

)x, ) sin(a), cos(a) ( y = (


This function is designed for a point that is expected to be rotating. In this function, the point’s

coordinate “x” and “y” are the designated to “sin(a), cos(a)”. The parameter “a” implies the

radian. Since sin + cos = 1, the function is equivalent to:(a)2 (a)2

+ = 1x2 y2

This function looks very familiar because it is a function of circle, where the sum of square x and

square y is equal to a definite value. According to the Pythagorean Theorem, the square root of

the constant “1” implies that the radius of the circle is also “1”. With these in mind, this drone

point must always be located at the position where its distance to (0,0) is exactly 1. With the

change of parameter “a”, the drone point to move along the path of the circle with radius 1. Thus,

by adjusting the parameter we can see a circular path a moving point.

Here are two examples showing how parameter “a” influences the position of point “A”, at

special radian in the circular path of A. (1.57 = , “a” is ranging from -5 to 5)2π

With the basic parameterized function, we can customize the motion of the point by adding more

coefficients and constants. For example, if we want to make the point moves in a circle with

radius 2 instead of 1, we will simply change the initial equation ) to:x, ) sin(a), cos(a) ( y = (

x, ) 2 in(a), 2 os(a)) ( y = ( × s × c

This one can be easily derived from the previous definition, sin + cos = 1. The general(a)2 (a)2

form is x sin + x cos = x, and radius equals x . We notice that the previous two× (a)2 × (a)2


coefficients are consistent with the radius, so by changing the coefficients of sines and cosines

simultaneously to the number of times we want, we will get the exact number radius as we want.

Note that this time, the radius of the circular of A changes to 2. Which twice the previous value.

This radius change is very useful. When later we are doing the Ferris Wheel, we need different

points on the shafts rotating in the same radian so that the Ferris Wheel will keep its basic look.

We will be able to assign different radius breaks to the shaft using this method.

System On Circular Path and Straight-Line Path

Up until now, the process is only viable for the motion of one single point. If we want to make a

series of points rotating or moving together, we have to adjust another feature. One simple way

to solve a system’s motion is to set the initial location of that series of points with respect to the

main point. The main point here has a parameterized function, and all the other points’ motion is

dependant on this single point. That means in whatever path the main points moves, all the others

points will follow it, keeping this particular shape intact.

There are two forms for the dependant points to follow the main point. The first form of which

the dependant points can follow the main point is keeping their radian difference a constant k. If

we initialize the main point according to the function , thex, ) r in(a), r os(a)) ( y = ( × s × c

parameterized function of the its dependant point will look like:


x, ) r in(a ), r os(a )) ( y = ( × s ± k × c ± k

Compare these two equations, notice that no matter what value “a” has, two sets of values inside

the parenthesis of the trigonometric function always has the difference of k. That means, the arc

created by the main point and the new point always has the radian k. Given a positive k, if the

content in the parenthesis is a + k, then the new point will appear at the right side of the main

point with a radian difference k. If it is a-k inside the parenthesis, the new point will appear at the

left side of the main point, with the same radian difference k. The following graphs shows this

system operation:

These two graphs shows the transformation of a system, which includes main point A and new

point B. The arc has radian , which is part of the circle + = 1. This is the main theory4π x2 y2

behind Ferris Wheel building. We call this circular moving.

Now, the second form of which a dependant point can follow a min point is adding constant j at

the end of each coordinate, outside the trigonometric function. Given the initialized main point

with coordinate , the parameterized function of the its dependantx, ) r in(a), r os(a)) ( y = ( × s × c

point will look like:

x, ) r in(a) , r os(a) ) ( y = ( × s ± j × c ± s

This function is very easy to understand. Adding or subtracting a self-defined constant from both

x and y coordinates will set the relative position at (j , s) regarding main point as its origin. The

value of j and s can be defined as needed. For instance, the following graph displays a main point


and new point in which the new point is always 0.5 unit higher and 0.5 unit right to the main

point.

Whichever value “a ” has, both the x, y coordinates of B is always 0.5 unit larger than that of A.

This is useful when we want the new point to stay at the same relative position of A. The best

demonstration of this method is the carts of the Ferris Wheel and the spines of the dragon. We

call this straight-line moving.

3.2 Time Parameter

To connect the computational model with reality, we need to incorporate the dimension of time.

The position of the drone, represented by parameterized coordinate function

is dependent on a. Since the increment of a is proportionalx, ) r in(a ), r os(a )) ( y = ( × s ± k × c ± k

to time t and the range of a can be represented in a linear oscillation. Generally, the linear

oscillation was shaped by a point going upward with some derivative and going downward with

the same derivative repeatedly :


Let’s break these curving down to one “V” shaped graph. In this case, the oscillation can be

completed by translating the V’s left vertex to its right vertex. By doing this over and over again,

we will be able to get the expected linear oscillation. In order to accomplish the exact translation,

we will introduce the parameter n,k,d. n is the number of times of which the “V” will translate, k

represents the length of line between the bottom vertex of “V” to either the right vertex or the left

vertex. Notice that k is always positive. By moving the “V” 2k distance leftward, the new “V”

would be located at the left side of old “V” while connecting itself to the old “V”. d represents

the little adjustment for the translation of the graph. The formula of the translation of “V” is:

t nk| − 2 − d|

Now let’s add two parameters a,b to adjust the slope of “V”. a is the upper bound of “V”, which

is also the highest position “V” can achieve. b is the lower bound of “V”, which is the

bottommost vertex of “V”. Both a and b are y values. The derivative of V will be .kb−a

Multiplying this derivative to the absolute value function, we can adjust the slope freely:

kb−a · t nk| − 2 − d|

The last thing is to make sure the bottommost vertex of “V” is always located at height of a.

Without this, we cannot really adjust the lower bound of the graph freely. This step is simple,

adding an “a” at the end of the graph will solve the problem. And thus we generated our final

formula of this linear oscillating.

=aparameter kb−a · t nk| − 2 − d| + a


3.3 Animation Video

To better demonstrate our light show, we combined all three of our animations into one

video and uploaded it to youtube under the link

https://www.youtube.com/watch?v=V9kknxh2Md4. Since we are not allowed to reveal

our name or any personal information in this document or related materials, we created a

new youtube account to publish this video. None of our information will be perceived by

viewing this video.

3.4 Ferris Wheel

Shaft

We start constructing our ferris wheel model by setting the center of the circle at the origin,

Each of the surrounding point on the shaft, excluding the cart, can be modeled by:0, ). A = ( 0

(x, ) d in(a ), os(a )) y = ( × s + r d × c + r

d: distance from the point to the center of the circle

r: angle formed by the line connecting the point and the center and the positive y axis in radians

https://www.youtube.com/watch?v=V9kknxh2Md4


For example, since the height or the diameter of the ferris wheel is 40m(mentioned in section

2.2), point O1, circled in green, has an coordinate function of

30 in(a .72), 30 os(a .72)) O1 = ( × s + 4 × c + 4

because the angle formed by the line AO1 and the positive y axis (highlighted in green).57 r = 4

is 270 degree or 4.72 radians.

Cart

Each square cart is modeled by a polygon function, of which one pivot point makes circular

motion arounds adjacent end point of one shaft while the other three points automatically adjust

their positions regarding the pivot point.

In the figure above, D1 is the pivot point and moves around C.

40 in(a .72), 40 os(a .72)) C = ( × s + 4 × c + 4

30 in(a .72), 30 os(a .72) ) O1 = ( × s + 4 × c + 4 − 2

Since D1 surrounds C instead of the origin, d stays the same while 2 is subtracted from the y

coordinate, representing the position and side length of the cart.


3.5 Dragon

Skeleton

We start our dragon animation model by outlining the skeleton (illustrated in red). Unlike the

ferris wheel in which all of the points are animated, only some regions of the dragon will be

moving. Thus, these moving joint points will have their positions generated as parameterized

functions while other static point remains as numerical coordinates.

It is important to note that the skeleton is used as a reference point for generating the actual

points on the outline of the image. It is deleted after the work is done and does not appear in the

final animation.

Regional Animation

There are three concurrent animations happening in the dragon’s mouth, head, and black, each

illustrated in a different color. On account of the different extents of circular motions in the three

regions, we designate three individual parameters to each of the regional animation. So instead of

using for all points, we have(x, ) d in(a ), os(a )) y = ( × s + r d × c + r


Mouth x, ) d in(a ), os(a )) ( y = ( × s + r d × c + r

Head x, ) d in(b ), os(b )) ( y = ( × s + r d × c + r

Back x, ) d in(c ), os(c )) ( y = ( × s + r d × c + r

In order to generate a more accurate and vivid animation, points of the outline of the image does

not always move around the center. Instead, they follow a finer path uniquely generated based on

the pivot points of their regions. In the figure below, the mouth region’s pivot point (in purple)

makes circular motion around the center of the image (in black) while the specific border point

(in blue) moves around the pivot. By doing so, each point can move along a different path,

making it possible to refine the animation.


3.6 Golden Gate Bridge

Since it will be strange to alter the shape of a static object like the golden gate bridge, we decide

to change its side instead to create a magnifying effect. Unlike the previous two displays, the

bridge animation does not use circular motion and is much more straightforward. All dots on the

image are pinpointed by hand and have the general coordinate of (x, y). In order to

proportionally magnify the bridge, we again apply parameterized function to make it

while a increases at a constant speed.x , ) ( * a y * a

3.7 Combining time and Coordinate Parameters

As we discuss in section 3.2, can be expressed as a function of time:aparameter

kb−a · t nk| − 2 − d| + a

Since we have each moving point’s pathway as a parameterized function, we can combine the

two to express the position of a specific point with regard of time.

For example, point O1 from section 3.4 has an coordinate of

30 in(a .72), 30 os(a .72))O1 = ( × s parameter + 4 × c parameter + 4


As we substitute the with the time function, O1 becomes:aparameter

30 in( ) .72), (30 os( ) .72)) O1 = ( × s kb−a · t nk| − 2 − d| + a + 4 × c k

b−a · t nk| − 2 − d| + a + 4

Where a,b are the lower and the upper bound of the , and k is the time period foraparameter

oscillation from a to b, which can be customized according preferences. N and d can be set based

to the original value of parameter. and the only variable is time. The same substitution process

can be applied to all 171 drones of our displays and the outcomes constitute a system of

equations which accurately give the position of each point on the coordinate system in any given

instant during the show, expressed by the generic equation:

oint x(t), (t)) P = ( y

3.8 Transformation

Contrary to the animations within each display, which does not encounter the intersection of

different drone’s flight paths, transformations between display need to avoid crashing of drones

while accounting for efficiency of rearrangement.

Let’s say display 1 just ends and is about to transform to display 2. Since the positions of all

drones within the three displays at any given time is known (section 3.7), we thus have the

coordinates of the drones at the display 1and the start of display 2. Using the pythagorean

theorem, we can calculate the distance between any two points from display 1 to display 2:

d = √(x ) y )2 − x12 + ( 2 − y1

2

We can generate a two-way table comparing all distances between points of the transformation.

A1 to An are points from display 1 and B1 to Bn are from display 2

B1 ... Bn

A1


...

An

Then we start from the top row A1, find the closest point from point A1 to display 2, and connect

the two. Repeat the same process for A2 and choose from the remaining n-1 points of display 2.

Since calculating distances will be both laborious and71 71 transformations 8482 1 × 1 × 2 = 5

tedious, we decide to illustrate our method with a simple transformation between a triangle and a

downward arrow.

I J K L M N O P

A 1.41 1 1 1.41 1 1 2 2

B 2 1 2.24 2.83 2.24 1 1.41 3.16

C 3.16 2.24 3.61 4.24 3.61 2.24 2 4.47

D 3.16 2.24 3 3.16 2.24 1 2.83 2.83

E 4.24 3.61 3.61 3.16 2.24 2.24 4.47 2

F 2.83 2.24 2.24 2 1 1 3.16 1.41

G 3 2 3.16 3.61 2.83 1.41 2.24 1.41


H 3.61 2.83 3.16 3 2 1.41 3.61 2.24

After connecting points between the two displays, we are able to generate a flight path diagram

which shows the intersections between trail paths(illustrated by yellow stars). In order to avoid

collision, to-be-intersecting drones will be moved to different planes (x and y stays the same, z

becomes different) to take their paths. Adjacent planes will be 0.5 meters apart to ensure safety.

3.9 Transformation pathway as a function of time

To express the transformation pathway as a function of time, we divide the drones into two

groups: z-plane-0 (stays on the initial plane) and z-plane-n (need to be moved to avoid collision).

For points staying in the same plane, since they are moving in a straight line, both x and y axis

has constant velocity, and their position can be expressed as

oint(x, ) x (T ) t, (T ) t) P y = ( 1 total − t + x2 y1 total − t + y2

Where is the total time for the point to move from to .T total , ) (x1 y1 , ) (x2 y2

can be calculated as Distance between and isT total . velocitydistance , ) (x1 y1 , ) (x2 y2

and maximum velocity of the drone is 3m/s. becomes √(x ) y )2 − x12 + ( 2 − y1

2 T total

.3√(x −x ) +(y −y )2 1

22 1

2

To conclude, the function to describe the position of z-plane-0 points is:

oint(x, ) x ( ) t, ( ) t) P y = ( 1 3√(x −x ) +(y −y )2 1

22 1

2

− t + x2 y1 3√(x −x ) +(y −y )2 1

22 1

2

− t + y2

Where are constants and t is the only variable., x , , y x1 2 y1 2

For z-plane-n points, their pathway can be described as a piecewise function, in which they move

Q-unit in the z direction:

, t < oint(x, , ) x , , t) P y z = ( 1 y1 3 3Q


, t > oint(x, , ) ( ) t, ( ) t, Q)P y z = x1 3√(x −x ) +(y −y )2 1

22 1

2

− t + x2 y1 3√(x −x ) +(y −y )2 1

22 1

2

− t + y2 3Q

3.10 Calculating Velocity during animations and transformation

As we are able to model the position of a drone as a function of time during both animation and

transformation,

oint(x, ) x(t), (t)) P y = ( y

The velocity at every instance can be calculated as the derivative of the x and y position function

elocity(x, ) x (t), (t)) V y = ( ′ y′

(t) V = √x (t) (t)′ 2 + y′ 2

During transformation when z-plane is involved, moving across the z-plane will be

1-dimensional motion and velocity is constant at 3m/s. After the drone travels to the designated

z-plane, its motion will again be 2-dimensional on the xy plane, of which the velocity can be

calculated using the function described above.

4. Requirements Specifications

4.1 Duration of the Show


FWA: Ferris Wheel Animation, FWTR: Ferris Wheel Transformation

DA: Dragon Animation, FTR: Dragons Transformation

GGWA: Gold gate bridge Animation

Since the highest point of the display is 120 meters from ground, time to launch the drones to the

ferris wheel image will be 120meter/3m/s = 40s. The animations of the ferris wheel involve a

half revolution of the image. The radius of the wheel is 20 meters. Distance travelled by the

drones on the circumference of the wheel is 62.8 meters, which takes 62.8/3 = 21s tor π =

complete. During the animation of the bridge, it is first magnified to two times the initial size

then shrink back. The gate expand from to0 meters(height) 0 meters(width) 2 × 6

The longest distance travelled by the drones is 120 meters (60m for both0 meters 20 meters. 4 × 1

expand and shrink) , which takes 40s to complete. Since the dragon animation involves

simultaneous movement of the three regions, a duration of 20 second to be sufficient to

demonstrate.

For transformation, though we do not have the accurate path of each single drones for our

display, we can assume the longest possible 2-dimensional path to be the height of our display

image --- 40m. Our transformation also involves the changing of plane to avoid collision. We

assume the largest number of intersecting path on our 171-drone-display is 50. The distance

between each plane is 0.5 meters. Therefore, the longest possible distance between the initial

plane and the destination plane is 25 meter. Moving to the plane and back is 50 meters, which is

added to the 40 meter 2-dimensional transforming path to generate a max transforming distance

of 90 meter. Therefore, it takes a maximum time of 30s to complete a transformation.

By the end of the bridge animation, the image ends up .0 meters(height) 0 meters(width) 2 × 6

The highest drone on the sky is 85.359m (lowest point of the display from section 2.2) + 20m =

105 meter, which takes 35 seconds to complete.

We add up the duration of launching, animations, transformations, and landing to get a total

duration of 215 seconds, which is about 3.5 minutes.


4.2 Safety Considerations

Under the renewed FAA(Federal Aviation Administration) policy regarding the flight height

limit of unmanned aircraft, the highest possible altitude a drone can reach is 400 feet, or 129

meters. We set the highest point of our display to be 120 meters from the ground, which is within

the safety limit. According to Intel, the developer of the Shooting StarTM drones which we will be

using for our light show, the maximum speed the drone could travel is 3 meters per second. We

use this speed limit for all our calculations. Since the shooting star drone has a dimension of

, we set the minimum distance between drones to be 0.5m to ensure.382m .382m .083m0 × 0 × 0

no adjacent drones will collide.

5. Final statement

On being tasked to model a light show for the city, we use ferris wheel, dragon, and golden gate

bridge as our display images. A total of 171 drones is used for the show, which requires 2400

m2 of launching and landing area and m3 of air space . We tilt our display plane to make450008

it vertical to the audience’s eyesight and avoid distortion. Animations of the displays are

modeled and simulated using geogebra. Using Parameterized functions, we are about to describe

the flight paths of the drones during animations and transformations with respect to time. We

also develop a model to efficiently transform drones from end of one display image to start of

another. The show has a duration of 3.5 minutes, which includes launching, animations,

transformations, and landing time. It is calculated using flight path and velocity of the drones.

Safety conditions, including the maximum velocity, minimum distance between drones, and fight

height limits are taken into consideration.


6. References

"Behind the Making of a Synchronized Drone Light Show." CBS News, CBS

Interactive, 27 Apr. 2017, www.cbsnews.com/news/

synchronized-drones-intel-light-show/. Accessed 20 Nov. 2017.

Cheung, Natalie. "Intel News Fact Sheet." Intel Newsroom, Intel,

newsroom.intel.com/wp-content/uploads/sites/11/2017/07/

Intel-Shooting-Star-Tech-Fact-Sheet-073117-1.pdf. Accessed 20 Nov. 2017.

---. "Technology behind the Intel Drone Light Shows." Robotics Tomorrow, LJB

Management, 9 May 2017, www.roboticstomorrow.com/article/2017/05/

technology-behind-the-intel-drone-light-shows/10022. Accessed 20 Nov. 2017.

Colalg. College Algebra Modules, colalg.math.csusb.edu/~devel/precalcdemo/param/

src/param.html. Accessed 20 Nov. 2017.

"FAA Doubles 'Blanket' Altitude for Many UAS Flights." FAA, Federal Aviation

Administration, 29 Mar. 2016, www.faa.gov/news/updates/?newsId=85264.

Accessed 20 Nov. 2017.

Feist, Jonathan. "Intel Shooting Star Version 2 and the Entertainment Business."

DroneRush, 7 Sept. 2017, www.dronerush.com/

intel-shooting-star-entertainment-business-10367/. Accessed 20 Nov. 2017.

"The Future of Drones Depends on Regulation, Not Just Technology." The

Economists, 10 June 2017, www.economist.com/news/technology-quarterly/

21723000-engineers-and-regulators-will-have-work-together-ensure-safety-drones-ta

ke. Accessed 20 Nov. 2017.

GeoGebra. International GeoGebra, www.geogebra.org/license/. Accessed 20 Nov. 2017.

Kaplan, Ken, editor. "Drones Light Night Sky to Set Record." IQ by Intel, Intel, 4 Nov. 2016,

iq.intel.com/500-drones-light-show-sets-record/. Accessed 20 Nov. 2017.


---, editor. "100 Dancing Drones Set World Record." IQ by Intel, Intel, 6 Jan.

2016, iq.intel.com/100-dancing-drones-set-world-record/. Accessed 20 Nov. 2017.

Locoma, Tyler. "Here’s Everything You Need to Know about the FAA’s Drone

Regulations." Digital Trends, Designtechnica, 3 Apr. 2017, www.digitaltrends.com/

cool-tech/drone-pilot-faa-regulations-guide/. Accessed 20 Nov. 2017.

Meola, Andrew. "The FAA Rules and Regulations You Need to Know to Keep Your

Drone Use Legal." Business Insider, 25 July 2017, www.businessinsider.com/

drones-law-faa-regulations-2017-7. Accessed 20 Nov. 2017.

Morris, Ian. "Intel Has Drones That Will Make Fireworks Obsolete." Forbes, 19

Sept. 2017, www.forbes.com/sites/ianmorris/2017/09/19/

intel-has-drones-that-will-make-fireworks-obsolete/#4927299e51db. Accessed

20 Nov. 2017.

SoftSchools.com. www.softschools.com/formulas/physics/velocity_formula/4/.

Accessed 20 Nov. 2017.

Walden, Josh. "Intel Announces Commercial Drone: Intel Falcon 8+ System."

Intel Newsroom, Intel, 11 Oct. 2016, newsroom.intel.com/editorials/

intel-announces-commercial-drone-intel-falcon-8-system/. Accessed 20 Nov. 2017.

Yarish, Gerry. "Mini Drones: Intel’s Shooting Star Drones Lit up the Sky during

the Super Bowl." Rotor Drone, Airage Media, www.rotordronemag.com/

inside-look-intels-shooting-star-drones-lit-sky-super-bowl/. Accessed 20 Nov. 2017.

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