Triangulation & Parallax

Unit Objective: Students will find indirect measurements using triangulation and parallax. Although the activities were written for a Trigonometry course, they could easily be adapted for other courses.

Activity 1: Distance Across a Canyon

Objective: Students will use triangulation to find the distance across a canyon.

Wyoming Standards:    Math 2.1-Students use similarity and perpendicularity to solve problems.
Math 2.2-Students communicate, using mathematical language, to analyze
properties and determine attributes of 2- and 3- dimensional objects.
Math 2.5-Students connect geometry with other mathematical topics.
Math 3.1-Students apply estimation and measurement using he appropriate
methods and units to solve problems involving length.
Math 3.3-Students identify and apply scale, ratios., and proportions in solving
measurement problems.
Math 3.5-Students solve indirect measurement problems.
Math 4.1-Students use algebraic concepts, symbols, and skills to represent and
solve real-world problems.
Sci 2.2-Students use inquiry to conduct scientific investigations.

Materials Needed: None

Time:  35 minutes

Directions: Make a sketch of a canyon and ask the students to suggest ways to measure the distance across. Make sure they understand that you cannot get to the other side. After the students discuss a few ideas introduce the concept of triangulation.

A person could make two measurements from different points along the canyon wall. Emphasize that the distance (S) between the points and the angles (A&B) could easily be measured. If we make sure angle A and angle B are equal, then the triangle formed would be isosceles. Since the triangle is isosceles the midpoint of S (also easy to find) could be connected with the point X on the opposite wall to form the altitude of the triangle.

Using only half of the triangle we would have a new right triangle. If we use Angle A, then (S/2) would be the adjacent side and d would be the opposite side. Using a tangent ratio we could solve for d:

Draw a small circle on the left side of the board and a point on the right side (the further away the better).

The circle represents the earth and the point represents a distant star. Ask the students if they believe we can use triangulation to measure the distance.

After some discussion point out that since the distance between the points (S) is so small compared to the distance d, the angles will be almost 90 degrees. For example if S=5 ft and d=10 miles, the angle would be 89.9946 degrees. Measuring angles like this is very difficult. A different method may be a better choice.

Activity 2: Explaining Parallax

Objective: The students will be introduced to parallax and how it can be used to find stellar distances.

Wyoming Standards:    Math 2.1-Students use similarity and perpendicularity to solve problems.
Math 2.2-Students communicate, using mathematical language, to analyze
properties and determine attributes of 2- and 3- dimensional objects.
Math 2.5-Students connect geometry with other mathematical topics.
Math 3.1-Students apply estimation and measurement using he appropriate
methods and units to solve problems involving length.
Math 3.3-Students identify and apply scale, ratios., and proportions in solving
measurement problems.
Math 3.5-Students solve indirect measurement problems.
Math 4.1-Students use algebraic concepts, symbols, and skills to represent and
solve real-world problems.
Sci 1.9-Students develop an understanding of scientific content through inquiry.
Sci 2.2-Students use inquiry to conduct scientific investigations.

Materials Needed: Computer projector, globe or other object

Time: 25 minutes

Directions:  Place the globe in the middle row on the front desk just at eye level for your students while seated. Place an X mark on the board directly in front of the globe also at eye level. Any students sitting in the same row as the globe should be able to see the globe but not the X.

Ask the student in the back of the row furthest right to describe how they see the globe in relation to the X. They should say something like “the globe is on the left and the X is on the right”. Make a quick sketch on the board.

Ask the student at the back of the middle row and the student at the back of the row furthest left to also describe what they see. Make a quick sketch of both of their views.

Explain that scientists describe this apparent shift as a “parallax shift” or just “parallax” for short. We often experience this in everyday life. Move the globe out of the way and have each student close their left eye, extend one arm, and cover the X mark with their thumb. Once the X is covered have the students open the left eye and close the right. The thumb will appear to move back and forth due to parallax. Parallax allows the brain to see two slightly different images (one from each eye) and combine them to produce a 3D image. Without this depth perception would be very difficult.

Describe how we can use parallax to measure distances. For our example we are going to measure the distance from our eyes to our thumb. Make a sketch showing how parallax can be found.

By knowing the distance between your eyes (2xR) and how much this shift measures in degrees (twice the parallax angle P) you can calculate the distance to your thumb (D). Finding the angle for this example is difficult but can be done. The formula that you would use is:

Tan (P)=R/D

We can use the same principal for measuring distances of far away objects. Next show the two parallax examples from the web sites.

Scroll to the bottom of the web page to “Continue to the Parallax Java Demo of Science” and then load the Java applet.

Click the 'Start' button. The large body in the middle could represent our sun. The two bodies moving back and forth would represent Venus and Mercury. The view shown would be a view from Earth if the earth was not moving but Venus and Mercury still moved. The reason the planets appear to move back and forth is because this view is from the side. This type of motion is called simple harmonic motion (SHM). A top view would have planets that appear to move in elliptical orbits. (Note: the web site does not use planets as their moving object, they use stars. We are using planets to simplify the model for the students).

First click the 'Animate' button and let the simulation run through a few cycles. Students should notice the near star appears to move back and forth in the image. They should also notice that the Earth is revolving around the sun. Emphasize that each revolution represents one year thus the apparent movement of the star is very slow. Students should also notice that during half of the cycle the image appears in the 'day'. That does not mean the star moves one direction at night and back during the day. It means that the star is in the daytime sky for half of the year and thus is not visible. It may be helpful to quickly start and stop the animation through one or two year cycles to emphasize these points.

Next click on 'Show Bounds' and run the animation. The extremes of the movement would be where the parallax angle would be measured. Stop the animation somewhere when the Earth is experiencing a daylight sky. Ask the students where the parallax angle is. Remind them the parallax angle is HALF of the angle formed between the red and blue lines.

Finally click on the 'Day/Night' button to turn off the daylight and run the animation. This is what the apparent motion of the near star would look like if the sun did not obscure the view. Again remind them this is a one year motion cycle, not one day.

Activity 3: Measuring Distance in Space

Objective: The students will use parallax to find indirect measurements of distances in space.

Wyoming Standards:    Math 2.1-Students use similarity and perpendicularity to solve problems.
Math 2.2-Students communicate, using mathematical language, to analyze
properties and determine attributes of 2- and 3- dimensional objects.
Math 2.5-Students connect geometry with other mathematical topics.
Math 3.1-Students apply estimation and measurement using he appropriate
methods and units to solve problems involving length.
Math 3.3-Students identify and apply scale, ratios., and proportions in solving
measurement problems.
Math 3.5-Students solve indirect measurement problems.
Math 4.1-Students use algebraic concepts, symbols, and skills to represent and
solve real-world problems.
Sci 1.9-Students develop an understanding of scientific content through inquiry.
Sci 2.2-Students use inquiry to conduct scientific investigations.

Materials Needed: Handouts from web site (Activity 1-Method 1)

Time: 35 minutes

Directions: (Note: We will closely follow Activity 1-Method 1 from the NASA-TRACEmerc web site. All handouts and images are available there. There would be no point in rewriting the directions when they were so well done by NASA originally).

Read the intro together and discuss the image. Students may ask how Mercury or the sun could be seen during half of the satellite's orbit. Point out that this satellite was in a polar orbit which runs perpendicular to the equator.

When working through the activity you can either give them a copy of all text from the web site including the explanation or just the data table and you as the instructor will have to do all of the explaining. I would give the students a copy of all the explanation and go through all of it step by step as a group.