Area of a circle trigonometry, uses the coarea formula. Define a function formula_10 by formula_11. Note ρ is a Lipschitz function whose gradient is a unit vector formula_12 (almost everywhere). Let "D" be the disc formula_13 in formula_14. We will show that formula_15, where formula_16 is the two-dimensional Lebesgue measure in formula_14. We shall assume that the one-dimensional Hausdorff measure of the circle formula_18 is formula_19, the circumference of the circle of radius "r". (This can be taken as the definition of circumference.) Then, by the coarea formula, Similar to the onion proof outlined above, we could exploit calculus in a different way in
Line Describing a Cone allowed him to actualize his ideas on the relationship between viewer and film and the medium of film itself. The thirty-minute film begins with a single white dot projected onto a black surface. As time progresses, the dot begins to form a curved line, tracing the circumference of a circle until the end of the line reaches its starting point. Meanwhile, particles in the air reveal the path of light in the space between the projector and the wall, making visible a cone of light. If the artist's display specifications are met, this beam of light projects between thirty and