Creating a visually appealing representation of a panda using mathematical functions is a fantastic project! Below, I’ve outlined a design plan along with the required mathematical equations. You can adjust the equations to fine-tune your design in Desmos.
Design Concept:
The concept is to create a stylized depiction of a panda, integrating various functions to represent different features such as the round body, eyes, and ears.
Equations for the Panda Design:
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Body (Ellipse) - A panda’s body can be modeled using a combination of quadratic functions:
- \((x^2 + (0.5y)^2 = 1)\) – elyptical shape scales vertically.
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Head (Ellipse):
- \((x^2 + (y - 2)^2 = 1)\) – creates a rounded head above the body.
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Eyes:
- Left eye (circle): \((x + 0.3)^2 + (y - 2.2)^2 = 0.1)\)
- Right eye (circle): \((x - 0.3)^2 + (y - 2.2)^2 = 0.1)\)
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Pupil:
- Left pupil (circle): \((x + 0.3)^2 + (y - 2.2)^2 = 0.03)\)
- Right pupil (circle): \((x - 0.3)^2 + (y - 2.2)^2 = 0.03)\)
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Ears (Semi-circles):
- Left ear: \(-\sqrt{1 - (x + 1)^2} + 2.5\) restricted to \([-1.5, -0.5]\)
- Right ear: \(-\sqrt{1 - (x - 1)^2} + 2.5\) restricted to \([-1.5, -0.5]\)
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Nose (Triangle):
- \(y = -2.5|x| + 2.1\) for a triangular shape restricted to \([-0.3, 0.3]\)
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Smiling Mouth (Quadratic function):
- \(y = -0.5(x^2 - 1) - 1\) – creates a smile, limited to \([-1, 1]\).
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Arms and Legs:
- Arms: \(y = -0.5|x| + 0\) for symmetry at both sides.
- Legs: \(y = -0.5|x| - 2\), representing the legs below the body.
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Tail (Circle):
- ((x+2)^2 + (y+0.5)^2 = 0.1)\ – creates a small tail.
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Background Environment (Exponential function) (Optional):
- A curve depicting bamboo or trees can add to the story. Use an exponential function: \(y = e^{-0.1x}\) for the background.
Implementation Steps in Desmos:
- Input each of the above equations into Desmos.
- Adjust the coefficients as needed to get the desired shape and size.
- Use brackets and carefully apply any transformations if necessary.
- Add vertical or horizontal guidelines if needed, but remember they don't count towards the 10-function requirement.
Reflection on Your Work:
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Inspirational Concept: I chose a panda because they symbolize peace and conservation efforts, which resonate with my interest in protecting nature.
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Function Selection: Each function was selected based on the specific geometric feature of the panda they represent, creatively combining various shapes to achieve a cohesive design.
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Challenges: The most daunting aspect was ensuring that the features looked proportional and recognizable. Iteration through feedback helped refine shapes and placements.
Presentation Outline:
- Describe the artistic intention behind the panda design.
- Explain how the linear function creates the arms, the quadratic function creates the mouth, and the exponential function can depict background elements.
- Reflect on the iterative process and collaboration that improved my design, focusing on mathematical techniques I feel more confident with.
Creativity and Final Reflection:
- If creating a new design, I might explore another animal or a fictional character using more intricate functions.
- I am most proud of how the panda’s facial expressions came together using simple functions; it made the design come alive.
- This project solidified my understanding of how math can shape art, revealing the depth and beauty of mathematical relationships.
Real-World Applications:
The skills from this project, such as graphing equations and understanding functions, are applicable in fields like game design, animation, and educational software development.
Advice for Future Projects:
Start with a clear concept, break down what each part of your design will entail, and be open to feedback. Iterating based on suggestions can greatly enhance your final product. Happy graphing!