Perspective techniques in the works of Leonardo da Vinci Automatic translate

Leonardo da Vinci developed a comprehensive system of perspective that surpassed the achievements of his contemporaries and laid the foundation for subsequent generations of artists. His approach combined mathematics, optics, and natural observations into a unified theoretical and practical system. The Florentine master didn’t simply apply existing rules; he created his own philosophy of depicting space, based on a profound study of the physiology of vision and the behavior of light.

Pyramid of vision and optical principles

Central to Leonardo’s theory is the concept of the visual pyramid — a geometric model explaining the process of perception. The artist defined perspective as a rational proof of how objects before the eye convey their image using a pyramid of lines. These lines emanate from the edges of each object’s surface, converge, and meet at a single point — the observer’s eye.

Leonardo described the atmosphere as filled with infinite pyramids composed of radiating straight lines. These pyramids are produced by the surfaces of objects in light and shadow. As they move away from an object, the pyramids become more acute. Although they intersect and crisscross as they spread, they never blend, but instead pass through the surrounding air independently.

The master applied the so-called pyramidal law to many processes. Light intensity decreases inversely proportional to the square of the distance between the light source and the object. Leonardo also used this principle in describing perspective, analyzing the attenuation of sound with distance, and the radiation of heat from a heated body.

Three-part perspective system

In his notebooks, Leonardo divided perspective into three branches, each responsible for a specific aspect of visual depth perception. The first branch — perspective of form — deals with the apparent decrease in size of opaque objects as they recede from the eye. This approach is also called linear perspective or diminishing perspective.

The second branch concerns the change in colors with distance from the eye. Leonardo described how colors lose saturation and change tone at greater distances. The third branch explains how objects should be rendered with less detail proportional to their distance. This classification remained unchanged in the master’s notes for approximately twenty years, demonstrating the stability of his theoretical views.

The artist distinguished between natural and artificial perspective. Natural perspective described the actual perception of space by the human eye, while artificial perspective focused on methods for reproducing this effect on a flat surface. Leonardo recognized that a scene projected according to the rules of one-point linear perspective never completely coincides with natural visual perspective.

Linear perspective in painting

Leonardo mastered the method of constructing one-point linear perspective, which he used thousands of times in his works. The most famous example is the fresco "The Last Supper," where the vanishing point is located directly behind Christ’s head. All parallel lines — the upper edges of the wall tapestries, the ceiling coffers — converge at this point, making the figure of Christ the compositional and semantic center of the work.

The proportions and spatial relationships in The Last Supper are carefully calculated to create a harmonious and balanced composition. This was in keeping with the Renaissance fascination with geometry and symmetry. Leonardo employed costruzione legittima — a legal method of construction — although historical evidence, such as sketches of the refectory, has not survived.

Round compass holes, used to mark perspective lines, were discovered on the walls. Double circular cuts were also found, used to outline the wreaths in three lunettes. Leonardo may have created a clay model before the composition on cartoons — this technique allowed him to pre-conceive the spatial solution.

In the "Annunciation," the artist demonstrates an impeccable mastery of foreshortening architectural elements. The marble table at which Mary is depicted is constructed with precise adherence to the laws of geometric perspective. However, Leonardo didn’t limit himself to the mechanical application of these rules — he made adjustments based on optical observations.

Aerial perspective and atmospheric effects

Leonardo made a decisive contribution to the development of aerial perspective — a technique for conveying depth through changes in color and clarity of distant objects. The artist was interested in how the air itself influences what we see. He believed this was due to water molecules and vapors in the atmosphere. According to the master’s observations, the higher one ascends, the bluer the air becomes, and the farther away an object, the bluer it appears.

In "The Virgin of the Rocks," aerial perspective creates a sense of expansive landscape space. Leonardo used his ingenious technique to convey the impression of a boundless landscape. He understood that we perceive distant objects less clearly and with altered color schemes. The mountains in the background transition from a lighter blue at their base to darker shades at the higher elevations. Details become blurred as they recede.

It’s no coincidence that Leonardo created an opening in the cave to make the river perfectly visible. This allowed him to demonstrate one of his greatest skills: the use of aerial perspective to create a sense of depth, enhancing the realism of the entire scene. The artist also employed this technique in the "Mona Lisa," where the smoky blue-gray tones of the distant mountains fade seamlessly into the horizon.

Leonardo observed that the illumination of an object depends on the angle of incidence. Light falling on a shadowed object at the most acute angle creates the highest illumination, while the darkest part receives light at an obtuse angle. Both light and shadow form pyramids. These observations formed the basis of his practical recommendations for depicting three-dimensional forms in space.

Sfumato as a tool of spatial depth

The sfumato technique became one of Leonardo’s greatest achievements in painting. The term comes from the Italian word "fumo," meaning smoke. The method creates soft, smoky transitions between tones and colors, eliminating hard edges. The artist asserted, "There are no edges in nature." He urged allowing the meaning of an object to become the object itself.

Sfumato is based on the finest oil glazes and patient blending. Leonardo applied multiple shades, creating depth through the subtle influence of underlying colors on the surface tones. He gradually reduced color saturation to create the illusion of distance or softness. The seamless blending mimics the natural diffusion of light.

In the "Mona Lisa," sfumato is evident in the model’s enigmatic smile and eyes. Leonardo used the technique to create a sense of depth and luminosity by blurring the lines and contours of the mouth and eyes. This softening effect, combined with the undefined shadows and subtle blurring of the edges, adds elusiveness to the enigmatic expression. The technique allows for subtle shifts in tone and shadow, heightening the sense of mystery.

Sfumato is closely related to aerial perspective. It’s a technique of softening edges that Leonardo favored. The artist gradually applied and blended numerous thin layers of paint on the canvas. He applied ultra-thin, transparent oil layers repeatedly, allowing him to achieve unprecedented realism in the rendering of skin, eyes, and atmosphere.

Degradation of shape and distortion

Leonardo developed the concept of deterioration of form — a complex of visual effects that cause the contours of objects to appear distorted in shape at a distance. This occurs to varying degrees and is associated with specific geometric and optical processes. Using basic axioms, the artist created a complex and highly accurate model of perspective, explaining how and why spatial objects appear as they do under a wide range of circumstances.

The master recognized that no scene projected according to the rules of one-point linear perspective completely matches natural visual perspective. Geometric discrepancies in the image are always apparent in one respect or another. This led Leonardo to introduce a fourth type of perspective — true dimensional perspective. It uses a flat perspective veil or window to capture images from the actual view.

This method is somewhat similar to images formed by linear perspective or the camera obscura. Brunelleschi may have used a similar method to develop and test the first geometrically constructed linear perspective images. Leonardo experimented with this technique in an attempt to overcome the limitations of traditionally constructed perspective.

In "The Adoration of the Magi," researchers discovered the use of multiple viewpoints simultaneously. Structures are straightened and projected according to only one viewpoint at a time. A range of scene variations is presented, taking into account non-rectangular structures. The scenes are generated using reverse perspective projections, raising discussion about the possible causes of these conscious or unconscious formal deviations.

Perspective of color and contrast

Leonardo’s second branch of perspective concerns the change in color with distance. The artist observed that colors lose saturation and contrast as they move away from the observer. These changes are due to the atmosphere between the eye and the observed objects. The first branch of perspective — the perspective of form — derives from the structure of the eye, while the other two are caused by atmospheric interference.

Leonardo recommended mixing paints with black to create realistic shadows. This was a change from the use of pure color in shadows, which created the vibrant but unnatural effect of the early Renaissance and Gothic periods. This theory of creating more realistic shadows coincided with the widespread adoption of perspective, leading to a trend toward more three-dimensional realism in painting.

In "Madonna of the Rocks," the artist demonstrates a subtle use of color gradations. The foreground is rendered in warm, rich tones with clear contrasts of light and shadow. The middle ground exhibits a reduction in color intensity and contrast. The background dissolves into cool, bluish hues with minimal tonal differences.

The artist explained that the illumination of a body depends on the angle of incidence of light, which influences color perception. He also took into account that light intensity varies inversely proportional to the square of the distance from the source. Leonardo translated these physical principles into practical recommendations for painters, creating a system that unites science and art.

The prospect of clarity and detail

The third branch of Leonardo’s perspective system concerns the reduction in clarity and detail of distant objects. The artist emphasized that objects should be less detailed in proportion to their distance. This observation is based on the optical properties of the atmosphere and the physiology of vision. As distance increases, the air diffuses light, resulting in a loss of sharpness of contours and fine details.

In "The Last Supper," the foreground, with the figures of the apostles, is rendered with the utmost detail. The folds of clothing, facial features, and hands are meticulously rendered. The table with its dishes and food is depicted with meticulous attention to texture and form. However, the background — the walls of the room and the landscape beyond the windows — is deliberately simplified. Details become less discernible, creating the illusion of spatial depth.

Leonardo observed that objects in the distance lose not only color and contrast but also the clarity of form. He described three aspects of perspective in painting: the decrease in size of objects with distance, the change in color as they move away from the eye, and the less detailed rendering of distant objects. All three aspects work together to create a convincing illusion of three-dimensional space on a flat surface.

In "Mona Lisa," the landscape behind the model exhibits a gradual loss of detail. Nearby elements — the terrace and the road — are relatively clear. The hills in the middle ground are depicted with less detail. The distant mountains become blurred, bluish silhouettes, devoid of any fine detail. This gradient of detail enhances the sense of spatial depth.

Chiaroscuro and form modeling

"The Virgin of the Rocks" showcases Leonardo’s revolutionary technique — the contrast of light and shadow, known as chiaroscuro. This method allows for the three-dimensional form of objects to be defined through tonal modeling. Although chiaroscuro differs from sfumato — which emphasizes smooth transitions without harsh contrasts — the two techniques complement each other in Leonardo’s system.

The artist understood that light and shadow form pyramids. Illuminated parts of an object receive light at acute angles, while shadowed areas are at obtuse angles to the light source. This geometric model helped Leonardo calculate the distribution of light and shadow on complex shapes. He applied these calculations to create convincing volume in figures.

In "The Virgin of the Rocks," the figures are modeled with exceptional subtlety. The light falls from above and to the left, creating soft gradations from illuminated areas to shadows. The faces and hands of the characters acquire a tangible sculptural quality thanks to carefully crafted tonal transitions. Leonardo avoids sharp boundaries between light and shadow, preferring a gradual fading.

The master recommended that painters carefully observe the behavior of light in nature. He urged them to study how objects are illuminated under various conditions — indoors, outdoors, in direct sunlight, and on cloudy days. These observations were to form the basis for depicting light and shadow in painting. Leonardo himself conducted systematic experiments, exploring optical phenomena.

Mathematical principles of composition

Leonardo used mathematical proportions in his compositions. In "The Last Supper," "Annunciation," and "Mona Lisa," researchers have discovered the use of the golden ratio, the golden spiral, and geometric shapes. Mathematics is widely embodied in Renaissance works, and artists of this era may have used mathematics to achieve more perfect composition and figure proportions.

In "The Virgin of the Rocks," the painting is composed from the bottom up, taking into account the golden rectangle. The intersections of the golden ratio occur at the eye of John the Baptist and the left eye of the angel. A circle drawn precisely from the top edge of the painting falls on the angel’s right eye. Leonardo connected the eyes of the figures in the central group through symbolic angles inscribed within the surrounding circle.

The eyes of the four figures serve as the basis for four triangles, each constructed according to symbolic angles. From the angel’s eye extends an angle of 108 degrees, pointing to the eye of the Christ Child and the intersection of the circle with the 72-degree line, representing the Madonna’s gaze upon Jesus. The angle of 108 degrees is directly related to the geometric symbolism of a regular pentagon.

The eyes are positioned along the circumference to form symbolic triangles: the yellow triangle with angles of 45, 90, and 45 degrees; the green with angles of 60, 45, and 75 degrees; and the blue with angles of 30, 60, and 90 degrees. The apex of the yellow triangle deviates 3.5 degrees from the Madonna’s right eye, but the triangle is too obvious to go unnoticed. The principle of superimposing triangles with symbolic angles is known from other paintings by Leonardo.

Promising experiments and limitations

Leonardo recognized the imperfections of linear perspective. His method of perspective was appropriate and significantly accurate, but it had flaws, and questions remained unresolved due to the limitations of the era. Nevertheless, his evolutionary approach to perspective created paintings that were unprecedented and unusual. His achievements proved influential and undeniable for subsequent generations.

The artist experimented with vision and perspective, applying the results of his research to painting. He used a documentary method of analysis to examine information from various sources, including explanations of his experiments and results. This helped readers gain a more comprehensive understanding of Leonardo’s ideas.

The master introduced a fourth type of perspective — true measured perspective — which uses a flat perspective veil or window to capture images from a real view. This method is somewhat similar to images formed by linear perspective or a camera obscura. A similar technique may have been used by Brunelleschi to develop and test the first geometrically constructed linear perspective images.

Leonardo understood the discrepancy between geometrically constructed perspective and natural vision. The eye sees the world not through a single vanishing point, but through binocular vision with two centers of projection. Furthermore, the natural field of view is curved, while linear perspective assumes a flat picture plane. Leonardo attempted to resolve these contradictions through compromises and adjustments.

The influence of anatomical research

Leonardo’s in-depth study of ocular anatomy influenced his theory of perspective. He based his study of perspective and optics on the functions of the eye. The artist also drew on ocular science in his astronomical research. In his manuscripts, he wrote: "The order of proof that the earth is a star is: first, identify the eye."

Leonardo conducted anatomical studies, collected in notebooks, where he described and depicted the tree-like, branching structures of biological systems, from the microcosm of an individual human organ to the macrocosm of the earth. His visualization of the complexity arising from tree growth was created during the same years in which he made fundamental observations about the rules of proportion that determine the branching structures of trees.

Influenced by Leonardo’s use of tonal modeling in sketches, medical images began to be depicted more realistically, creating three dimensions. This had never been shown in anatomical drawings before. His sketches not only depicted images but also included detailed captions, demonstrating his medical knowledge. Leonardo’s discoveries had a profound impact on the development of physiology, as new knowledge about the human body was uncovered through his observational drawings.

Practical application of the theory

Leonardo created literally thousands of perspective drawings in his notebooks. Many of them utilized the key method of linear perspective. The artist applied his theory not only to painting but also to architectural projects. Manuscript B contains several drawings of centrally planned churches, some of which are presented in plan and paired with a bird’s-eye view.

Using a bird’s-eye view instead of an elevated one represented an innovative depiction technique. It combined the immediacy of a perspective view with the measurability of the façades, capturing the buildings’ three-dimensionality. To understand the rationale behind this choice, researchers used parametric modeling to digitally reconstruct the churches depicted in Manuscript B.

When working on cartoons and preparatory drawings, Leonardo used various tools to accurately construct perspective. A compass, a ruler, and strings — all of these helped the artist transfer mathematical calculations to the plane. Traces of compasses and strings used to mark perspective lines have been found in frescoes.

Leonardo recommended that artists create perspective screens — transparent grids through which nature could be observed and what was seen could be transferred to a flat surface. This method of true, measured perspective allowed for capturing real-life views with great precision. Although the technique required time and patience, it produced results close to natural vision.

Integration of science and art

Leonardo united art and science in the Renaissance through the geometric features of his work. Using geometry, he imbued his works with a unique sense of order and beauty through the golden ratio and perspective. In his creative process, Leonardo applied mathematical principles and demonstrated a profound ability to observe the natural world.

This interdisciplinary fusion not only enhanced artistic expression but also reflected modern society’s desire for an integrated system of knowledge. The relationship between art and mathematics has been discussed many times in the past, but rarely has a comprehensive analysis been undertaken of the application of various mathematical methods to works of art. Mathematics and art have always influenced each other throughout history — in form and thought.

Leonardo developed complex theories containing definitions, principles, and explanations that are remarkably modern. His approach to perspective was a rational argument applied to how objects before the eye convey their image. Using a pyramid of lines, the outer edge of which begins at the edges of each object’s surface, converges, and meets at a single point — the eye.

Treatise on Painting

Leonardo’s notes on perspective were collected in various manuscripts, later compiled into the "Treatise on Painting." This masterpiece of art theory explores the principles of perspective, composition, and anatomy in painting. The work emphasizes observation of nature, technical mastery, and the importance of light and shadow, offering profound insights into the creative process.

The outline of the painting book included three sections on perspective. The first section covers the reasons for the apparent reduction of objects as they recede from the eye, known as diminishing perspective. The second describes how colors change as they recede from the eye. The third and final section explains how objects should appear less complete in proportion to their distance.

Leonardo wrote, "The first thing in painting is that the objects represented should appear in relief, and the backgrounds surrounding them at various distances should appear in the vertical plane of the foreground of the picture through the three branches of perspective." These three classes of perspective — the reduction of the clarity of the forms of objects, the reduction of their size, and the reduction of color — work together.

The first of these results from the structure of the eye, while the other two are caused by the atmosphere between the eye and the objects being observed. The second essential thing in painting is appropriate action and proper variety in figures, so that people don’t all look like brothers. These guidelines indicate the scheme Leonardo sought to implement in organizing his studies of perspective as applied to painting.

The legacy of a promising system

Leonardo’s achievements in perspective influenced subsequent generations of artists. The Leonardeschi, the master’s followers, adopted the technique of aerial perspective he introduced. The use of sfumato to create smooth transitions and atmospheric depth became a characteristic feature of many 16th-century painters. Art historians note that this technique is absent from the works of some artists of the same period, such as Raphael, although he adopted the use of sfumato.

Leonardo’s method, combining the mathematical precision of linear perspective with the optical observations of aerial perspective and the tonal modeling of sfumato, created an unprecedented system for depicting space. This system transcended the mechanical application of geometric rules, taking into account the physiology of vision and the physical properties of light and atmosphere.

Modern research using eye-tracking technology confirms the effectiveness of Leonardo’s compositional techniques. Experiments recording participants’ eye movements while viewing paintings demonstrate that perspective does indeed direct viewer attention. While the effect may not be as straightforward as previously assumed, perspective structure does influence the perception of an image.

The use of mathematical methods in works of art is widely used by contemporary artists, with abstract and cubist artists employing mathematical methods most prominently. Mathematics and art continue to influence each other. The profound meaning of artistic works can also be conveyed to the viewer through mathematical language.

Leonardo da Vinci’s system of perspective combined geometry, optics, and the physiology of vision into a coherent artistic methodology. His tripartite division into the perspectives of form, color, and clarity provided artists with comprehensive tools for creating a convincing illusion of three-dimensional space. The introduction of the concepts of aerial perspective, sfumato, and degraded form expanded the possibilities of painting, allowing it to convey not only geometric depth but also atmospheric effects, optical phenomena, and the subtleties of natural perception.