Convergence of Light and Shadow: the 5/8 Penumbra
Geometry of luminous and shadow convergence, measuring how much of a light each region sees
A sheet of shadow-geometry, crowded with crossing rays and light-source circles casting triangular shadows. Leonardo distinguishes the 'luminous convergence' from the 'shadow convergence' and notes that shadows cast by a light larger than the body differ at their edges from their middles. In a lettered analysis he treats all shadow lines as straight, since they converge among straight light lines, and computes that the regions t S and l S each receive 5/8 of the whole light—seen directly from part of the source and obliquely from the rest—so that the triangle t X p holds less light and the areas outside the shadows exactly 5/8. Further untranscribed figures cover the page.
On this page
Edges of a shadow cast by a large light
The edges of the width of shadows cast by lights larger than the shadowy bodies differ from their middles, as though they arose from a varying quality of light.
Luminous convergence and shadow convergence
A figure is captioned with the terms Shadow, Light, Luminous convergence, Shadow convergence, Luminous body, naming the distinct zones formed where rays cross.
Regions lit by 5/8 of the light source
t S is more luminous than the rest of the middle triangle because it is seen directly from n o and obliquely from o g and c f—together 5/8 of the whole light; l S likewise receives 5/8, so the triangle t X p holds less light and the areas outside the shadows exactly 5/8.
Shadow lines are straight, converging among straight light lines
All the lines of the shadows are straight, because they converge between light lines that are also straight—the premise for the lettered triangles t, S, l, X, p that measure the illumination.