Steiner surface. It was during a trip to Rome in 1844 that Jakob Steiner first discovered the fourth-degree surface that today bears his name; for this reason it is sometimes referred to as the Roman surface. Each of its tangent planes has the characteristic property that it intersects the surface in a pair of conics. The Steiner surface also contains three double lines that meet one another in a triple point. Steiner never published these and other findings concerning the surface. A colleague, Karl Weierstrass, first published a paper on the surface and Steiner's results in 1863, the year of Steiner's death.
Jakob Steiner (born March 18, 1796, Utzenstorf, Switzerland—died April 1, 1863, Bern) was a Swiss mathematician who was one of the founders of modern synthetic and projective geometry.
As the son of a small farmer, Steiner had no early schooling and did not learn to write until he was 14. Against the wishes of his parents, at 18 he entered the Pestalozzi School at Yverdon, Switzerland, where his extraordinary geometric intuition was discovered. Later he went to the University of Heidelberg and the University of Berlin to study, supporting himself precariously as a tutor. By 1824 he had studied the geometric transformations that led him to the theory of inversive geometry, but he did not publish this work. The founding in 1826 of the first regular publication devoted to mathematics, Crelle’s Journal, gave Steiner an opportunity to publish some of his other original geometric discoveries. In 1832 he received an honorary doctorate from the University of Königsberg, and two years later he occupied the chair of geometry established for him at Berlin, a post he held until his death.
What is the Steiner surface?The Steiner surface (or Roman surface), discovered in 1844 by Swiss mathematician Jakob Steiner. Each of its tangent planes intersects the surface in a pair of conics. The Steiner surface also contains three double lines that meet one another in a triple point.
During his lifetime some considered Steiner the greatest geometer since Apollonius of Perga (c. 262–190 bce), and his works on synthetic geometry were considered authoritative. He had an extreme dislike for the use of algebra and analysis, and he often expressed the opinion that calculation hampered thinking, whereas pure geometry stimulated creative thought. By the end of the century, however, it was generally recognized that Karl von Staudt (1798–1867), who worked in relative isolation at the University of Erlangen, had made far deeper contributions to a systematic theory of pure geometry. Nevertheless, Steiner contributed many basic concepts and results in projective geometry. For example, during a trip to Rome in 1844 he discovered a transformation of the real projective plane (the set of lines through the origin in ordinary three-dimensional space) that maps each line of the projective plane to one point on the Steiner surface (also known as the Roman surface). Steiner never published these and other findings concerning the surface. A colleague, Karl Weierstrass, first published a paper on the surface and Steiner’s results in 1863, the year of Steiner’s death. Steiner’s other work was primarily on the properties of algebraic curves and surfaces and on the solution of isoperimetric problems. His collected writings were published posthumously as Gesammelte Werke, 2 vol. (1881–82; “Collected Works”).
Projective drawingThe sight lines drawn from the image in the reality plane (RP) to the artist's eye intersect the picture plane (PP) to form a projective, or perspective, drawing. The horizontal line drawn parallel to PP corresponds to the horizon. Early perspective experimenters sometimes used translucent paper or glass for the picture plane, which they drew on while looking through a small hole to keep their focus steady.
projective geometry, branch of mathematics that deals with the relationships between geometric figures and the images, or mappings, that result from projecting them onto another surface. Common examples of projections are the shadows cast by opaque objects and motion pictures displayed on a screen.
Projective geometry has its origins in the early Italian Renaissance, particularly in the architectural drawings of Filippo Brunelleschi (1377–1446) and Leon Battista Alberti (1404–72), who invented the method of perspective drawing. By this method, as shown in the figure, the eye of the painter is connected to points on the landscape (the horizontal reality plane, RP) by so-called sight lines. The intersection of these sight lines with the vertical picture plane (PP) generates the drawing. Thus, the reality plane is projected onto the picture plane, hence the name projectivegeometry. See alsogeometry: Linear perspective.
Although some isolated properties concerning projections were known in antiquity, particularly in the study of optics, it was not until the 17th century that mathematicians returned to the subject. The French mathematicians Girard Desargues (1591–1661) and Blaise Pascal (1623–62) took the first significant steps by examining what properties of figures were preserved (or invariant) under perspective mappings. The subject’s real importance, however, became clear only after 1800 in the works of several other French mathematicians, notably Jean-Victor Poncelet (1788–1867). In general, by ignoring geometric measurements such as distances and angles, projective geometry enables a clearer understanding of some more generic properties of geometric objects. Such insights have since been incorporated in many more advanced areas of mathematics.
Fundamental theorem of similarityThe formula in this diagram reads k is to l as m is to n if and only if line DE is parallel to line AB. This theorem then enables one to show that the small and large triangles are similar.
A theorem from Euclid’s Elements (c. 300 bc) states that if a line is drawn through a triangle such that it is parallel to one side (see the figure), then the line will divide the other two sides proportionately; that is, the ratio of segments on each side will be equal. This is known as the proportional segments theorem, or the fundamental theorem of similarity, and for triangle ABC, shown in the diagram, with line segment DE parallel to side AB, the theorem corresponds to the mathematical expression CD/DA = CE/EB.
Now consider the effect produced by projecting these line segments onto another plane as shown in the figure. The first thing to note is that the projected line segments A′B′ and D′E′ are not parallel; i.e., angles are not preserved. From the point of view of the projection, the parallel lines AB and DE appear to converge at the horizon, or at infinity, whose projection in the picture plane is labeled Ω. (It was Desargues who first introduced a single point at infinity to represent the projected intersection of parallel lines. Furthermore, he collected all the points along the horizon in one line at infinity.) With the introduction of Ω, the projected figure corresponds to a theorem discovered by Menelaus of Alexandria in the 1st century ad:
C′D′/D′A′ = C′E′/E′B′ ∙ ΩB′/ΩA′.
Since the factor ΩB′/ΩA′ corrects for the projective distortion in lengths, Menelaus’s theorem can be seen as a projective variant of the proportional segments theorem.
With Desargues’s provision of infinitely distant points for parallels, the reality plane and the projective plane are essentially interchangeable—that is, ignoring distances and directions (angles), which are not preserved in the projection. Other properties are preserved, however. For instance, two different points have a unique connecting line, and two different lines have a unique point of intersection. Although almost nothing else seems to be invariant under projective mappings, one should note that lines are mapped onto lines. This means that if three points are collinear (share a common line), then the same will be true for their projections. Thus, collinearity is another invariant property. Similarly, if three lines meet in a common point, so will their projections.
The following theorem is of fundamental importance for projective geometry. In its first variant, by Pappus of Alexandria (fl. ad 320) as shown in the figure, it only uses collinearity:
Are you a student?
Get a special academic rate on Britannica Premium.
Let the distinct points A, B, C and D, E, F be on two different lines. Then the three intersection points—x of AE and BD, y of AF and CD, and z of BF and CE—are collinear.
There is one more important invariant under projective mappings, known as the cross ratio (see the figure). Given four distinct collinear points A, B, C, and D, the cross ratio is defined as
CRat(A, B, C, D) = AC/BC ∙ BD/AD.
It may also be written as the quotient of two ratios:
CRat(A, B, C, D) = AC/BC : AD/BD.
The latter formulation reveals the cross ratio as a ratio of ratios of distances. And while neither distance nor the ratio of distance is preserved under projection, Pappus first proved the startling fact that the cross ratio was invariant—that is,
CRat(A, B, C, D) = CRat(A′, B′, C′, D′).
However, this result remained a mere curiosity until its real significance became gradually clear in the 19th century as mappings became more and more important for transforming problems from one mathematical domain to another.
Feedback
Thank you for your feedback
Our editors will review what you’ve submitted and determine whether to revise the article.
verifiedCite
While every effort has been made to follow citation style rules, there may be some discrepancies.
Please refer to the appropriate style manual or other sources if you have any questions.
Select Citation Style
The Editors of Encyclopaedia Britannica. "Jakob Steiner". Encyclopedia Britannica, 28 Mar. 2025, https://www.britannica.com/biography/Jakob-Steiner. Accessed 29 April 2025.
Our editors will review what you’ve submitted and determine whether to revise the article.
print
Print
Please select which sections you would like to print:
verifiedCite
While every effort has been made to follow citation style rules, there may be some discrepancies.
Please refer to the appropriate style manual or other sources if you have any questions.
