Drag r. Watch A(r) trace live. dA/dr and d²A/dr² tell you where area is momentarily flat, and whether that flat point is a peak or a floor.
Circlegrowing
r = 3.00
ΔV = — (fixed)
drag to orbit
x=6.00, y=4.00, z=2.00
ΔV = — (fixed)
drag to orbit
3.00 units
x (width) = 6.00
y (height) = 3.00
camera → Δx strip = 1.0× (moves camera, shape unchanged)
camera → Δy strip = 1.0× (moves camera, shape unchanged)
x = 6.00
y = 4.00
z = 2.00
camera → Δx face = 1.0× (moves camera, x unchanged)
camera → Δy face = 1.0× (moves camera, y unchanged)
camera → Δz face = 1.0× (moves camera, z unchanged)
Link:
a =1.0b =-1.0
C =0.0
camera zoom = 1.0×
A(r) tracedA/dr > 0
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Derivatives at this instant
Area A
28.27
π · r²
dA/dr — exact rate of growth
18.85
2π · r (limit as Δr → 0)
ΔA — actual strip volume (Δr fixed)
—
A(r+Δr) − A(r) vs. dA/dr · Δr
d²A/dr² — curvature
6.28
2π (constant)
Critical point (dA/dr = 0)
r = 0
only solution to 2πr = 0
Why A = πr²? — cut into sectors, unfold into a rectangle
12 sector-pairs
unfold = 0.00
As the number of sectors grows, the zigzag straightens into a rectangle: base = half the circumference = πr, height = r.
Area = base × height = πr · r = πr² — exactly the circle's area, since cutting and rearranging never changes total area.