Isometric / Orthographic Drawing Project
Orthographic top, Side, and Front
My Anamorphic 3-D Drawing Project
Original Image
An anamorphic drawing is an image that is project from a single point to create a drawing that appears “warped” from all angles other than that of which the image was projected. This can create an interesting affect when used in combination with 3D shapes, giving them a standout property to the eye, but only from a single point. This is a characteristic unique to anamorphic drawings, and has lead to their creation and implementation amongst a variety of art mediums.
The materials used to complete this project are:
- 1 sheet of white poster board for drawing
- 1 laminate transparency
- 1 Vis-a-Vis marker to draw shape on laminate transparency
- scotch tape to secure transparency to shoe box
- shoe box to be used as a stand for drawing
- sharpened pencil(s)
- strong eraser(s)
- yard stick
- sharpie(s)
- shading pencil(s)
Our anamorphic drawing is a result of a projection because we plotted points on our poster board after looking through our original drawing and having one group member line up points on the poster board to corresponding points on our transparency. Because the transparency was at an angle to the poster board the “projections” or vision of one group member created points that varied based on perspective, and this created our projected anamorphic drawing.
Our most relevant challenge during this process was plotting the points with accuracy and refinement. Because there were variables that were hard to control (like height+direction of eyesight) we had some points that were out of place, but after drawing our initial shape out, we were able to refine our lines and visualize what we wanted our final version to look like.
The materials used to complete this project are:
- 1 sheet of white poster board for drawing
- 1 laminate transparency
- 1 Vis-a-Vis marker to draw shape on laminate transparency
- scotch tape to secure transparency to shoe box
- shoe box to be used as a stand for drawing
- sharpened pencil(s)
- strong eraser(s)
- yard stick
- sharpie(s)
- shading pencil(s)
Our anamorphic drawing is a result of a projection because we plotted points on our poster board after looking through our original drawing and having one group member line up points on the poster board to corresponding points on our transparency. Because the transparency was at an angle to the poster board the “projections” or vision of one group member created points that varied based on perspective, and this created our projected anamorphic drawing.
Our most relevant challenge during this process was plotting the points with accuracy and refinement. Because there were variables that were hard to control (like height+direction of eyesight) we had some points that were out of place, but after drawing our initial shape out, we were able to refine our lines and visualize what we wanted our final version to look like.
South Pole
We found the height of the object by using the equation h=tanΘ ∗ x. For the height of the we used South Pole we used the equation h=tan11˚ ∗ 220, which gave us the answer h=42.76 ft.
West Lightpost
We found the height of the object by using the equation h=tanΘ ∗ x. For the height of the we used West Lightpost we used the equation h=tan20˚ ∗ 55, which gave us the answer h=20.01 ft.
East Tree
We found the height of the object by using the equation h=tanΘ ∗ x. For the height of the we used East Tree we used the equation h=tan21˚ ∗ 17.56, which gave us the answer h=6.74 ft.
Tables for Locations
Inscribed Shapes In a Circle
Hexaflexagon
In my hexaflexagon project, I used both rotational symmetry and line reflection. When I was creating my design, I reflected different patterns to create a visually appealing overall design. As you can see in the pictures below, some of the shapes and patterns I created are able to reflect across a line onto each other, and some of the designs contain rotational symmetry when the hexaflexagon was folded.
One design feature that I am proudest of on my hexaflexagon is the use of simple shapes with varying colors throughout the hexaflexagon. The simple lines and shapes made the reflections and rotations easier to identify, and created a visual pleasing image when you rotate the hexaflexagon. If I were to make any refinements to my piece now that I understand the line-reflection and rotational symmetry concepts, I would add more shapes and be precise when creating the lines on my hexaflexagon. I believe that if I implemented these two ideas my hexaflexagon would be even more visually pleasing than before. One thing that I learned about myself from this activity is that I am a very visual learner. I learn best when I'm able to work with and create hands on projects such as the hexaflexagon. This hands on learning gave me a clearer understanding of reflections and rotational symmetry, and I feel I benefited from this activity greatly.
GeoGebra 'Graffiti' Lab
To complete this lab, we used concepts on rotation and reflection. As you may notice in the visual piece of the lab, the different colored parts are reflections or rotations of each other. These shapes contain line symmetry, and all are able to rotate around a point located in the center. Sometimes parts of the shape contain both reflections and rotations in the same translation.
During the process of creating this lab I learned that, as I mentioned in the 'Hexaflexagon' section, sometimes I learn best through visuals and creation of concepts. For me, this was the first GeoGebra lab we've done where we were able to bring geometry concepts to life in colorful and interactive designs. Although my learning curve on GeoGebra hasn't been lightning fast, this project warmed me up to the idea of conceptualizing ideas through technology, and opened a new perspective for me on learning by looking.
Two Rivers GeoGebra Lab
There is a sewage treatment plant at the point where two rivers meet. You want to build a house near the two rivers (upstream from the
sewage plant, naturally), but you want the house to be at least 5 miles from the sewage plant. You visit each of the rivers to go fishing about the same number of times but being lazy, you want to minimize the amount of walking you do. You want the sum of the distances from your house to the two rivers to be minimal, that is, the smallest distance. In GeoGebra we created a sketch following these instructions. The below images are screenshots of the labs.
sewage plant, naturally), but you want the house to be at least 5 miles from the sewage plant. You visit each of the rivers to go fishing about the same number of times but being lazy, you want to minimize the amount of walking you do. You want the sum of the distances from your house to the two rivers to be minimal, that is, the smallest distance. In GeoGebra we created a sketch following these instructions. The below images are screenshots of the labs.
The scenario pictured above would not be correct because the distance from the House to point A on the West river plus the distance from the House to point B on the East river is not equivalent to the shortest possible distance. This location satisfies the requirement that the house must be out of the sewage zone, but does not meet the requirement for the shortest possible path.
The scenario pictured above would be the correct answer because the distance from the House to point A on the West river plus the distance from the House to point B on the East creates the shortest possible distance. This location satisfies the requirement that the house must be out of the sewage zone, and also meets the requirement for the shortest possible path.
The Burning Tent GeoGebra Lab
A camper out for a hike is returning to his campsite. The shortest distance between him and his campsite is along a straight line, but as he approaches his campsite, he sees that her tent is on fire! He must run to the river to fill a bucket of water, and then run to his tent to put out the blazing flames. What is the shortest path he can take? In this exploration we investigated the minimal two-part path that goes from a point to a line and then to another point. This lab explored concepts learned from earlier GeoGebra labs using point > line > point ideas, but used a real example that solidified the ideas for me.
This scenario would not work because the segment between the camper and River does not lie on top of the segment between the Camper and TentFire', which is important when determining shortest distance. Also, the incoming and outgoing angles are not equivalent, which is another indicator that the distance travelled would not be the shortest in this scenario.
This scenario would work because the segment between the camper and River lies on top of the segment between the Camper and TentFire', which indicates that point River does not need to be moved on line AB. In this instance, the incoming and outgoing angles are equivalent, which is another indicator that the distance travelled would indeed be the shortest distance in this scenario.