Download Overlapping Photographs Meeting

Overlapping Photographs Meeting
provided by the Association for Unmanned Vehicle
Systems International (AUVSI) Foundation
www.auvsifoundation.org
Topic
This meeting allows your club members to investigate an application of basic trigonometry. Afterward, club
members can test their own ability to take overlapping photographs that can be put together to create a
larger picture.
Materials Needed
♦♦ Copies of the Overlapping Photographs Activity Sheet (downloaded from www.mathcounts.org)
♦♦ Digital camera (or another way of taking photographs) - optional at club meeting
♦♦ Photo printer - optional at club meeting
Meeting Plan
This meeting plan introduces your students to the concept of taking overlapping photographs to construct
a larger picture. The Overlapping Photographs Activity Sheet will demonstrate a scientific application of
taking overlapping photographs, and then students can engage in a more artistic application of the process.
Getting Started
Pass out the Overlapping
Photographs Activity Sheet (shown
here and available for download
from www.mathcounts.org), and
allow students to work as a group
on the problem. If your students
are not familiar with using the basic
trigonometric functions with right
triangles, you will have to provide
more assistance through Step 2
of the solution. However, the math
skills necessary (but perhaps not the
vocabulary) for Steps 3-5 should be
accessible to most middle school
students. Complete solutions for the
steps are available for download at
www.mathcounts.org.
Discussion
Why would taking photographs with
overlap be important in the scenario
described on the worksheet?
Overlapping Photographs Activity Sheet
(provided by the Association for Unmanned Vehicle Systems International Foundation)
The Scenario
An unmanned aerial vehicle flies over a target at a speed of 10 meters/second from an altitude of
100 meters. The aircraft is capable of taking photographs with a field of view of 4 degrees.
The Problem
What should be the camera frame rate, in frames per second, to enable a 50% overlap of photos
(along the direction of travel) taken of the ground target area? Express your answer as a decimal to
the nearest tenth.
The Solution
Step 1: We need to understand what a “field view of 4 degrees” tells us. In this
figure, the 4 degrees is shown. The area on the ground between the two arrows will
be captured in a photograph.
2
2
100
x
L
4
meters Step 2: Do we know how wide (L) the area covered in each
______________
photograph is, in meters? Remember we also know the vehicle is at an altitude
of 100 meters. In trigonometry, a length can easily be calculated if it is part of a
right triangle and we know one other side length of the right triangle. Note that
the 4 degree angle can be split into two 2 degree angles, forming
two congruent, right triangles. We can now use the trigonometric
function tan(θ) = x/y to solve for x, and therefore, L. How wide, in
meters, is the area covered in each photograph? Express your
answer as a decimal to the nearest hundredth.
y
x
______________ Step 3: The scenario requires a 50% overlap in the photo images. This means they
wish to take a photo at a point where 50% of the previous photo is in the next photo taken. Immediately
after a photo is taken, what ratio of the total width of the photo area, L, does the plane need to fly to take
the next photo? Express your answer as a common fraction.
______________
seconds Step 4: Velocity (or the speed) is defined as V = ΔX/Δt where V = velocity,
ΔX = distance traveled and Δt = the time to travel the distance ΔX. If ΔX is the distance the vehicle must
There are many reasons for wanting
travel between photos and the velocity is the velocity provided in the scenario, what is the time between
successive photos, in seconds? Express your answer as a decimal to the nearest hundredth.
overlap. Fifty percent is more than
typical, but it allows for redundancy
______________ Step 5: Frame rate is defined as the number of photos per second. Using the
information from Step 4, what is the frame rate for this scenario? Express your answer as a decimal to the
in image recovery and verification of
nearest tenth.
aircraft velocity. What would happen
if the plane was flying faster than we
expected and we did not allow for overlap? How can we determine aircraft velocity from the images?
photos/sec
2011–2012 MATHCOUNTS Club Resource Guide
35
Next Steps
Ask students (1) to take a series of photographs that has at least 20% overlap from picture to picture,
(2) to print out the photographs and (3) to piece them together to create one larger picture.
Some forms of art make use of this process, and there are software programs that will blend multiple
pictures into one large photo as long as there is at least 25% overlap. Similarly, 360º virtual tours are
created by “stitching” together a series of photographs.
When someone takes a series of photographs to combine into one large picture or to use to make a
360º virtual tour, what factors are important to ensure that the final product looks as close to “real life” as
possible?
www.auvsifoundation.org
36 2011–2012 MATHCOUNTS Club Resource Guide
Overlapping Photographs Activity Sheet
(provided by the Association for Unmanned Vehicle Systems International Foundation)
The Scenario
An unmanned aerial vehicle flies over a target at a speed of 10 meters/second from an altitude of
100 meters. The aircraft is capable of taking photographs with a field of view of 4 degrees.
The Problem
What should be the camera frame rate, in frames per second, to enable a 50% overlap of photos
(along the direction of travel) taken of the ground target area? Express your answer as a decimal to
the nearest tenth.
The Solution
Step 1: We need to understand what a “field view of 4 degrees” tells us. In this
figure, the 4 degrees is shown. The area on the ground between the two arrows will
be captured in a photograph.
2
2
100
L
x
4
meters Step 2: Do we know how wide (L) the area covered in each
______________
photograph is, in meters? Remember we also know the vehicle is at an altitude
of 100 meters. In trigonometry, a length can easily be calculated if it is part of a
right triangle and we know one other side length of the right triangle. Note that
the 4 degree angle can be split into two 2 degree angles, forming
two congruent, right triangles. We can now use the trigonometric
function tan(θ) = x/y to solve for x, and therefore, L. How wide, in
meters, is the area covered in each photograph? Express your
answer as a decimal to the nearest hundredth.
y
x
______________ Step 3: The scenario requires a 50% overlap in the photo images. This means they
wish to take a photo at a point where 50% of the previous photo is in the next photo taken. Immediately
after a photo is taken, what ratio of the total width of the photo area, L, does the plane need to fly to take
the next photo? Express your answer as a common fraction.
______________
seconds Step 4: Velocity (or the speed) is defined as V = ΔX/Δt where V = velocity,
ΔX = distance traveled and Δt = the time to travel the distance ΔX. If ΔX is the distance the vehicle must
travel between photos and the velocity is the velocity provided in the scenario, what is the time between
successive photos, in seconds? Express your answer as a decimal to the nearest hundredth.
photos/sec Step 5: Frame rate is defined as the number of photos per second. Using the
______________
information from Step 4, what is the frame rate for this scenario? Express your answer as a decimal to the
nearest tenth.
Overlapping Photographs Activity Sheet Answer Key
(provided by the Association for Unmanned Vehicle Systems International Foundation)
The Scenario
An unmanned aerial vehicle flies over a target at a speed of 10 meters/second from an altitude of
100 meters. The aircraft is capable of taking photographs with a field of view of 4 degrees.
The Problem
What should be the camera frame rate, in frames per second, to enable a 50% overlap of photos
(along the direction of travel) taken of the ground target area? Express your answer as a decimal to
the nearest tenth.
The Solution
Step 1: We need to understand what a “field view of 4 degrees” tells us. In this figure,
the 4 degrees is shown. The area on the ground between the two arrows will be captured
in a photograph.
4
meters Step 2: Do we know how wide (L) the area covered in each
______________
photograph is, in meters? Remember we also know the vehicle is at an altitude of 100 meters. In trigonometry, a length can easily be calculated if it is part
2 2
of a right triangle and we know one other side length of the right triangle.
100
Note that the 4 degree angle can be split into two 2 degree angles, forming
x
L
y
two congruent, right triangles. We can now use the trigonometric function
tan(θ) = x/y to solve for x, and therefore, L. How wide, in meters, is the area covered in each
photograph? Express your answer as a decimal to the nearest hundredth.
x
Tan θ = x/y, so x = y(Tan θ ) = 100(0.0349207695) = 3.492 meters,
and L = 6.98, to the nearest hundredth.
______________ Step 3: The scenario requires a 50% overlap in the photo images. This means they
wish to take a photo at a point where 50% of the previous photo is in the next photo taken. Immediately
after a photo is taken, what ratio of the total width of the photo area, L, does the plane need to fly to take
the next photo? Express your answer as a common fraction.
The plane needs to fly (1/2)L between
photos. Note: This is 3.49 meters.
seconds Step 4: Velocity (or the speed) is defined as V = ΔX/Δt where V = velocity,
______________
ΔX = distance traveled and Δt = the time to travel the distance ΔX. If ΔX is the distance the vehicle must
travel between photos and the velocity is the velocity provided in the scenario, what is the time between
successive photos, in seconds? Express your answer as a decimal to the nearest hundredth.
V = ΔX/Δt; Δt = ΔX/V; Δt = (3.49 meters)/(10 meters/sec); Δt = 0.35, to the nearest hundredth.
photos/sec Step 5: Frame rate is defined as the number of photos per second. Using the
______________
information from Step 4, what is the frame rate for this scenario? Express your answer as a decimal to the
nearest tenth.
(1 photo / 0.35 seconds) = 2.9 photos per second.