Process Paper: Polar Equation Project

Animation

Problem 1:

What is the most efficient method to animate a flower composed of polar curves?

Solution:

To solve this problem, I realized that the usage of a slider minimizes work but also provides freedom with the control of the curves. I decided to use two sliders — one for random motion (to make things look more “alive”), and one for the animation itself.

Sliders:

• “s” slider: as “s” increases from 0.25 to 1, the flower blooms.
• “v” slider: periodic motion, fluctuates within the 0–1 range. Actual variable used is “u.”
• When “s” increases to 1, “u” will equal the value of “v” determined by the slider.
• When “s” is below 1, the value of s80(and therefore “u”) will almost always be negligible as it is almost zero. This allows for periodic motion when the flower is fully bloomed.

Problem 2:

What if I want certain graphs to only show up in certain frames? Is there a way to control when graphs are visible?

Solution:

• I decided that in order to make certain graphs only show up in certain frames, I must make the graphs based off of “s”. To solve this, I added a s-term within these functions that would be imaginary or real based on the value of s, rendering the graph unplottable at certain frames. Suppose you have a curve with the form r=f(). In addition, you also have a function q = s-n. It can be seen that the variable q will only be a real number (and graphable on the xy-plane) when s>n. Therefore, having a curve in the form r= f() + q (where is some value arbitrarily close to 0), would be only be visible when s>n.

Aesthetics

Problem 1:

Solid fills are incompatible with polar curves.

Solution:

• The approach that I decided to have to this problem was to derive general formulas of polar/rectangular curves in their polar/rectangular counterparts, which would allow for solid fills.

Polar Equations:

• Polar equation (2nd) of a translated conic (1st):

• Rectangular equation (2nd & 4th) of a rose curve (1st & 3rd):

• Polar equation (2nd) of a line (1st):

Problem 2:

A flower enlarging consistently looks robotic and unlively. Is there a way to make it look more realistic?

Solution:

• To create different the growth of the layers in the flower, I decided that using different scaling factors for each equation such as s2, s3, s allows for more control over the “growth rate” of each curve. In addition, adding a constant term within the sin/cos inside the rose curve allows different layers of the flower to have different starting rotations, making the flower look more random and nature-like. Adding a term within the sin/cos in the rose curve that fluctuates will allow the petals to rotate with respect to the pole. These tiny fluctuations make the flower look more realistic. Adjusting the constant term inside of the rose curve that multiplies the angle allows me to control the amount of petals that I would like each “layer” of the flower to have. Lastly, usage of the “rgb” function makes each “layer” of the flower look more vibrant.
• With rgb (left), without rgb (right).

Problem 3:

The color of a flower petal should not stay consistent over time.

Solution:

• To solve this problem, I decided that adding the “s” variable to the “rgb” function would allow for the color to change over time as the slider increases. Adding an “s” with a plus sign makes the color seem brighter, while “s” with a minus sign makes the color seem darker as time progresses. Adding a non-constant multiplier to “s” within the “rgb” function allows for me to adjust the rate of brightening/darkening of a certain layer.

Plotting Log

(vast majority of work done on 3/25, small tweaks done on 3/26 through 4/6)

• Created a black solid background.
• Created colors using the “rgb” functions.
• Created stem, made with shades of green.
• Created polar curves. Used more petals in the center, less on the outside. Added a coefficient to each of the equations to modify the maximum “radius” of the curve.
• Used dotted lines in the middle to simulate pollen inside the flower.

Tweaks:

• Simplified variables within the equations.
• Adjusted colors of certain elements of the flower.
• Adjusted the size of the stem & flower petals.
• Adjusted the rate of expansion for each of the flower petals.