Here is Carpaccio’s painting of Saint George and the Dragon.
Add red lines from the left corner at 75, 60, 45, 30, and 15 degrees off horizontal. Note that the steepest one, at left, is an axis around which the dragon’s tail oscillates. The vertical dropped from the top of the 60-degree line passes through the right corner of the square-planned tower in the middle ground. Crucially, the vertical dropped from the top of the 30-degree line passes through the point where the lance intersects its crosspiece.
Take the lower left corner as the center of a very large octagon whose right-hand facet coincides with the red line dropped from the top of the 45-degree line. A line of slope 22.5 degrees from the lower left corner of the painting intersects the red vertical dropped from the 30-degree red line, thus locating the precise point of the lance/crosspiece intersection.
Taking that intersection point on the lance as a center, create two dodecagons; one whose lower facet aligns with the bottom of the painting, and a smaller one whose upper facet aligns with the top of the painting. The lance, with its 15-degree slope, passes through corners of these dodecagons. The tree foliage at the upper right also lies along this axis.
The right margin of the painting may be found by dropping a 45-degree diagonal down from the uppermost left corner of the small dodecagon down to the bottom margin of the painting; this operation thus sets the overall aspect ratio of the composition. A line of 15-degree slope descending to the left from this same upper point passes along the shadows on the rocky hills about a third of the way across the composition. A horizontal through the lance intersection point aligns with the ground level of the hills on the far right.
This painting is the property of the Scuola di San Giorgio degli Schiavoni, Venice.
This analysis was based on a color-enhanced version of the following image:
Here is the Visitation scene painted by Piero di Cosimo.
The inner field of the painting is a perfect square, flanked by narrow strips on the bottom and sides. The center of the square locates the clasping hands of the Virgin and Elizabeth.
The center of the square also defines a level, identified here with the orange horizontal, that describes the edge of the architectural platforms on which the small figures in the middle ground stand. From the ends of that horizontal, orange lines of 30-degree slope are now launched.
Then, from the points where those orange lines intersect the top of the panel, yellow verticals descend, which frame a yellow circle, in which a yellow square can be inscribed in a quadrature operation. Note that the bottom facet of the square aligns with the back of the platform on which the Virgin and Elizabeth stand.
The front of the platform corresponds to the bottom tip of a green equilateral triangle whose baseline corresponds to the equator of the square. A similar triangle above this equator helps to locate the heads of the two women as they lean towards each other, with their wimples sloped at 60 degrees from the horizontal. Their heads and the Virgin’s halo fit into the almond-shaped figure that circumscribes the paired triangles. One odd detail about this painting is that the horizon line of the seascape in the deep background between the women is not actually horizontal. Instead, it falls gently from left to right.
It is tempting to imagine that this might have been because the artist connected the wrong lines in his geometrical armature. Here, blue lines of 30-degree slope depart from the equator of the panel, within the red frame, and one can find the skewed horizon line quite precisely by connecting the point at left where the rising blue line intersects the yellow vertical with the point at right where the falling blue line intersects the side of the green triangle.
This painting is the property of the National Gallery of Art in Washington, DC.
This analysis was based on the image:
That image has now been superseded by:
Here is an anonymous Italian Renaissance painting of an ideal city painting now preserved at the Walters Gallery in Baltimore. Its width is considerably greater than its height.
More specifically, its width is 2.828 times its height, or two times the square root of two. This can be seen by unfolding the diagonals of two squares set on its outer edges. Note that the right margin of the left square aligns with the central axis of the amphitheater in the distance.
Several key elements in the painting can be found by tracing lines of 30-degree and 60-degree slope upward from its outer lower corners. The right margin of the amphitheater aligns with the vertical where the 30-degree line cuts the left arc, and the left margin can be found by reflection. The baptistery at right fits into a nearly equivalent frame, although it is compressed slightly to the right.
The front corners of the palaces in the foreground can be found by first drawing a yellow horizontal through the point where the 30-degree orange line cuts the axis of the amphitheater, and then drawing yellow verticals through the circled points halfway between the intersections of this horizontal with the 45- and 60-degree lines.
The rear corners of the palaces are exactly halfway between the panel margins and its center, as the green verticals indicate, and the green horizon line can be found at the level where the rising green lines of 30-degree slope cut the orange margins of the amphitheater and the baptistery.
The top edge of the right-hand palace can be found by running a blue line from the vanishing point to the top of the yellow vertical describing the right-hand palace corner. From the point where this blue line cuts the orange line of 60-degree slope, a blue vertical can be dropped to find the axis of the column in the foreground, and a similar construction also works in the left half of the panel, even though the palace there is shorter.
The perspectival lines in the pavement can be found by connecting the vanishing point to already constructed points in the foreground, and a similar scheme located at the front edge of the platforms on which the palaces stand.
This painting is the property of the Walters Art Museum in Baltimore, Maryland.
This analysis was based on the image: