Up and Down (1947)
The canonical mature curvilinear lithograph. A courtyard seen simultaneously from above and below through a shared central vanishing point — the impossible spatial reading depends on Escher's intuitive curvilinear projection.
Perspective · Curvilinear projection · 180° field of view · Advanced
5-Point Fisheye Perspective
The perspective for when linear breaks down. Five vanishing points — one at the centre of vision plus four at the cardinal edges of the visual field — project a full 180-degree hemisphere onto a flat circle. Straight lines curve as great-circle arcs; horizons bow; verticals fan inward. The system that Escher worked out intuitively in Up and Down (1947), that Flocon and Barre formalised mathematically in La perspective curviligne (1968), and that Dick Termes extended to spherical canvases starting in the 1970s. The visual language of immersive VR previews, panoramic concept art, and any scene wider than a human eye can see at once.
- Mathematical foundation
- Hauck (1870s) → Flocon & Barre (1968)
- First artistic mature use
- 1947 (Escher, Up and Down)
- Vanishing points
- 5 (centre + 4 edges)
- Field of view
- up to 180° (full hemisphere)
- Difficulty
- Advanced
- Also known as
- Curvilinear, fisheye, 5VP
See the curvilinear projection on five wide-angle subjects
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An interior in 5-point projection: the room's straight walls become arcs curving from each cardinal-edge vanishing point toward the centre. The horizon arc bows visibly. Vertical edges fan from top to bottom of the projected circle.
What the overlay shows
The 5-point fisheye overlay draws a circle representing the 180-degree visual hemisphere, with five vanishing points: one at the centre of the circle (where lines straight ahead converge) and four on the perimeter at the top, bottom, left, and right (where lines at 90 degrees from straight-ahead converge). Curved guide arcs radiate from each pair of opposed perimeter VPs through the centre, showing how parallel lines in three-dimensional space project as great-circle arcs.
The overlay is parameterised by circle size and visual-angle coverage. A full 180-degree hemisphere uses the entire circle; partial fisheye projections (typical photographic fisheye lenses cover 120 to 180 degrees) use partial coverage of the circle and produce less extreme curvature. The projection itself can use equidistant, stereographic, or orthographic mapping depending on the artist's intent and the camera reference being matched.
The projection geometry
5-point fisheye perspective projects three-dimensional space onto a hemispherical surface (visualised as a circular flat canvas representing the unfolded hemisphere). The mathematics is non-trivial:
visual hemisphere = circle of radius r
centre VP = direction straight ahead (azimuth 0°)
4 edge VPs = directions ±90° (left, right, up, down)
parallel lines project as great-circle arcs
each great-circle arc meets at the appropriate edge VP
specific projection mapping (equidistant, stereographic, orthographic) determines arc curvature
Flocon and Barre's La perspective curviligne (1968) is the canonical mathematical treatment, developing the system from projective geometry first principles and demonstrating the construction methods practically.1 The 1987 English translation by Robert Hansen made the work accessible to anglophone artists. Dick Termes's six-point variant adds a sixth vanishing point at directly-behind-the-viewer, completing the 360-degree spherical projection rather than the 180-degree hemispherical projection that 5-point covers.2
History — from Hauck to Escher to Termes
Verified history (with primary sources)
1870s — Hauck. Guido Hauck, a German mathematician, publishes early work on subjective perspective and optical projection that anticipates the curvilinear approach.3 Hauck's writing remains largely untranslated and influences the German-language perspective tradition more than the broader Western art world.
1947 — Escher, Up and Down. M.C. Escher creates the lithograph Up and Down, showing a courtyard seen simultaneously from above and below through a shared central vanishing point with curved verticals.4 Escher worked the construction out by hand using intuition and engineering drawings, without the formal mathematics that would not be published for another two decades.
1956 — Escher, Print Gallery. Escher creates one of his most mathematically complex prints: a young man in a gallery looking at a print of a town that contains the gallery he is standing in. The continuous spiral distortion approximates extreme curvilinear projection. The central "hole" in the print was solved in 2003 by Hendrik Lenstra of Leiden University using Riemann surface theory.4
1968 — Flocon and Barre. Albert Flocon and André Barre publish La perspective curviligne through Flammarion in Paris, the first complete mathematical and pedagogical treatment of curvilinear perspective for artists.1 The book covers four-point, five-point, and six-point variants and demonstrates construction methods.
1970s — Termes. Dick Termes develops his six-point spherical projection system at his studio in Spearfish, South Dakota.2 Termespheres — paintings on spherical canvases that read correctly from any viewing angle — become the canonical demonstration of full-sphere projection. Termes's own writings document the construction method.
1987 — Hansen translation. Robert Hansen translates Flocon and Barre into English as Curvilinear Perspective: From Visual Space to the Constructed Image, published by the University of California Press.1 The translation makes curvilinear theory available to a broad anglophone artistic audience for the first time.
2013 — Robertson and Bertling. Scott Robertson and Thomas Bertling publish How to Draw through Design Studio Press, including substantial treatment of curvilinear perspective for contemporary concept-art practice.5 The book popularises 5-point fisheye in concept-art training programmes.
Unverified claims that won't die
"Fisheye perspective started with fisheye lenses." Curvilinear projection in art predates photographic fisheye lenses. Escher's Up and Down (1947) precedes the first commercial fisheye lens (Nikon Fisheye-Nikkor 8mm, released 1962) by fifteen years. The artistic and photographic traditions developed in parallel from a shared theoretical foundation in projective geometry.
"Curvilinear perspective is just stylised distortion." It is a mathematically valid projection of three-dimensional space onto a curved surface. Linear perspective is the special case where the projection surface is flat; curvilinear perspective is the general case. Calling one "correct" and the other "distortion" is choosing one projection convention over another — both are equally valid representations of the same underlying geometry.
When to use it (and when not)
| If you want to... | Use 5-point fisheye | Don't use it for... | Difficulty |
|---|---|---|---|
| Represent a full 180-degree hemisphere of view | Linear perspective cannot do this without infinite stretching at the edges | Standard architectural drawings within a 60-degree visual angle (use 2-point) | Advanced |
| Match a fisheye-photographic reference | The drawing convention corresponds to one variant of photographic fisheye projection | Standard rectilinear-photo references (use 2- or 3-point) | Advanced |
| VR / 360-degree environment design or previs | VR scenes inherently use curvilinear projection on the headset's spherical canvas | Mobile or desktop 2D interfaces (no projection needed) | Advanced |
| Create Escher-style ambiguous-space images | The continuous distortion enables impossible-architecture compositions | Conventional landscape or architectural rendering | Advanced |
| Concept art for immersive panoramic scenes | Cinematic ultra-wide cityscapes, fantasy panoramas, alien landscape vistas | Standard scene composition where the audience is not at a hero-vista vantage point | Advanced |
Famous applications
Six demonstrations of curvilinear perspective in art and design.
Print Gallery (1956)
Continuous spiral distortion approximating extreme curvilinear projection. Hendrik Lenstra of Leiden University used Riemann surface theory in 2003 to solve the central blank "hole" that Escher left unfinished.
Termespheres (1970s–present)
The six-point spherical extension of 5-point projection. Spherical canvases painted to read correctly from any viewing angle. The most thoroughly worked-out demonstration of curvilinear perspective theory in fine art.
VR previsualisation pipelines
VR scene composition inherently uses curvilinear projection because the head-mounted display surface is hemispherical relative to the viewer. Curvilinear previs sketches match what the headset will render at runtime.
Fisheye photography (1962–present)
Photographic fisheye lenses produce curvilinear images matching 5-point projection geometry. Sports photography, skateboarding, and music-video work have established fisheye photography as a recognised visual genre.
Concept art panoramas (2010s–present)
Hero-shot environment concept art for open-world games increasingly uses 5-point fisheye for "the impossible vista" — cinematic scenes that exceed any human field of view. Scott Robertson's curriculum at Art Center taught the technique to most of this generation.
Common mistakes
1
Inconsistent arc curvature
The most common error. Arcs at different distances from centre have different curvatures — edges near the perimeter curve more than edges near the centre. Beginners draw all arcs at the same curvature, producing geometrically invalid projections that look "off" without the artist being able to identify why.
Fix: use a flexible curve template or a string-and-pencil great-circle construction. Each arc's curvature scales with distance from centre; the overlay shows the correct curvature at each position.
2
Using 5-point when 2- or 3-point would work
Curvilinear perspective looks dramatic and beginners reach for it reflexively. For typical scenes within a 60-degree visual angle, linear perspective handles the geometry cleanly without the extra construction complexity. Using 5-point on subjects that don't require it produces curves that read as gratuitous stylisation.
Fix: use 5-point only when the visual angle exceeds approximately 90 degrees, or when you specifically want the curvilinear stylistic effect. For typical architectural and figure work, 2-point is correct.
3
Treating the circle as decorative rather than structural
The circle in 5-point projection is the boundary of the visible hemisphere, not a decorative frame. Beginners sometimes draw a curvilinear scene and then crop a rectangular composition out of the middle, throwing away the geometric integrity that the full circle provides.
Fix: compose within the circle. The boundary is part of the work. If you need a rectangular crop, use a perspective system designed for rectangular output (1-, 2-, or 3-point) instead.
4
Mixing curvilinear and linear elements
Some beginners draw the central architecture in linear perspective and add curvilinear elements at the edges of the scene. The result reads as two different optical realities adjacent to each other — visually incoherent unless deliberately stylised for ambiguous-space effect (Escher's territory).
Fix: commit to one projection system per work. If you need curvilinear at the edges, draw the entire scene curvilinear including the centre. The Escher-style mixed projection is an advanced technique requiring deliberate planning.
How different disciplines use it
For illustrators
Editorial and book illustrators use curvilinear perspective for immersive scenes — sci-fi vistas, fantasy panoramas, ambiguous-space dream sequences. The Escher-style ambiguous-space tradition remains the canonical artistic reference for serious curvilinear work.
For photographers
Photographic fisheye lenses produce curvilinear images directly. Sports, action, music video, and skateboarding photography have established fisheye as a recognised visual genre. Architectural photographers occasionally use fisheye for compressed-space interior shots where conventional lenses would not fit the full subject.
For concept artists
5-point fisheye is the standard technique for "the impossible vista" — hero-shot panoramic scenes in concept art for AAA games and films. Scott Robertson's curriculum at the Art Center College of Design taught the method to most of the current generation of working game concept artists; Robertson and Bertling's How to Draw (2013) brought it to a broader market.
For VR and immersive designers
VR previsualisation sketches inherently use curvilinear projection because the headset surface is hemispherical relative to the viewer. Working in 5-point fisheye for previs sketches matches what the headset will render at runtime, avoiding the disconnect between linear-projection previs and curvilinear-projection final output.
"Linear perspective is the special case. Curvilinear perspective is the general case. The Renaissance chose the special case because it was easier to construct on flat panel paintings, and we have been pretending that flat is normal ever since. Curvilinear perspective is what the eye actually sees."
Albert Flocon and André Barre, La perspective curviligne, trans. Hansen (UC Press 1987)1
Frequently asked questions
What is 5-point fisheye perspective?
A curvilinear perspective system with five vanishing points — one at the centre of vision plus four at the cardinal edges of the visual field — used to project a full 180-degree hemispherical view onto a flat circular canvas. Straight architectural lines become curved arcs.
Who developed curvilinear perspective?
Mathematical foundation in nineteenth-century projective geometry (Hauck 1870s). Intuitive artistic use by M.C. Escher in Up and Down (1947). First systematic treatise: Albert Flocon and André Barre's La perspective curviligne (1968, English trans. Hansen 1987). Dick Termes extended to six-point spherical from the 1970s.
Why does linear perspective break down at wide angles?
Linear perspective assumes a flat picture plane. At wide visual angles, objects near the picture-plane edges project with extreme stretching because the angle between picture plane and viewer's line-of-sight becomes shallow.
Is fisheye perspective the same as fisheye photography?
Closely related. Photographic fisheye lenses produce images matching one variant of curvilinear projection. The 5-point fisheye drawing method produces visually similar results manually.
When should I use 5-point instead of linear perspective?
When your visual angle exceeds approximately 90 degrees, when representing a full hemisphere of vision, or for curvilinear stylistic effect (immersive panoramas, Escher-style spaces, fisheye-photographic mood).
Who is Dick Termes?
American artist (b. 1941) based in Spearfish, South Dakota. Developed the six-point spherical projection system for painting on spherical canvases (termespheres) starting in the 1970s. The most thorough working-out of curvilinear perspective in fine art.
Did Escher use 5-point perspective?
Escher used curvilinear projection intuitively in Up and Down (1947), Print Gallery (1956), and others without naming it. He worked out each construction by hand without the formal Flocon-Barre theory that postdates most of his major curvilinear work.
What is the difference between 5-point and 4-point perspective?
Both are curvilinear systems. Four-point perspective uses four vanishing points and is typically drawn on a vertical strip to wrap a panoramic view left-to-right while keeping verticals straight — useful for cylindrical panoramas. Five-point perspective adds a fifth vanishing point at the centre of vision and arranges all five around and within a circle, which lets it project a full 180-degree hemisphere (the fisheye look) where horizons curve and verticals fan inward. In short: 4-point handles wide horizontal panoramas; 5-point handles the full hemispherical fisheye.
When should I use curvilinear perspective?
Reach for curvilinear (4-, 5-, or 6-point) when the visual angle exceeds roughly 90 degrees, when you need to show a full hemisphere or panorama in one frame, or when you want the bubble-perspective look as a deliberate stylistic choice. Within a 60-degree visual angle, linear 1-, 2-, or 3-point perspective is simpler and produces less distortion.
Where can I learn 5-point in depth?
Flocon and Barre's Curvilinear Perspective (UC Press 1987 English translation) is the canonical theoretical treatment. Scott Robertson and Thomas Bertling's How to Draw (Design Studio Press 2013) is the most accessible contemporary artist's introduction.
References
- Flocon, A. & Barre, A. Curvilinear Perspective: From Visual Space to the Constructed Image. Trans. R. Hansen, University of California Press (1987). ISBN 0-520-05927-5. Original French: La perspective curviligne, Flammarion, Paris (1968). The canonical mathematical treatment.
- Termes, D. New Perspective Systems. Termes Studio, Spearfish, SD (1998). Plus monographs and exhibition catalogues from the 1980s onward documenting the six-point termesphere projection.
- Hauck, G. Die subjektive Perspektive und die horizontalen Curvaturen des dorischen Stils. Verlag der K.B. Akademie, Stuttgart (1879). Early German-language treatment of subjective and curvilinear perspective.
- Bool, F.H. et al. (eds.) M.C. Escher: His Life and Complete Graphic Work. Harry N. Abrams (1992). ISBN 0-8109-8113-9. Catalogue raisonné documenting Escher's intuitive curvilinear constructions including Up and Down (1947) and Print Gallery (1956).
- Robertson, S. & Bertling, T. How to Draw: Drawing and Sketching Objects and Environments from Your Imagination. Design Studio Press (2013). ISBN 978-1-933492-72-7. Contemporary artist's introduction to 5-point fisheye in concept-art practice.
- Lenstra, H. "Escher and the Droste effect." Notices of the AMS 50(4), 446–451 (2003). Mathematical solution of the central blank space in Escher's Print Gallery using Riemann surfaces.
- Veltman, K.H. Linear Perspective and the Visual Dimensions of Science and Art. Deutscher Kunstverlag (1986). ISBN 3-422-00779-X. Scholarly survey including the curvilinear tradition.
- Hansen, R. "This Curving World: Hyperbolic Linear Perspective." Journal of Aesthetics and Art Criticism 32(2), 147–161 (1973). DOI: 10.2307/429038. Hansen's earlier theoretical paper that prepared his Flocon-Barre translation.
Notes from the studio · Three practitioners on curvilinear perspective
Illustrative composites of how the tool gets used in practice — not quotes from named individuals.
For VR previs the curvilinear projection matches what the headset will render. Sketching in 5-point instead of linear saves a complete iteration of "looks good on the panel, wrong in the headset."
I use curvilinear for hero-shot panoramas in game environment art. Wide-angle without the eye-distortion of pushing linear perspective beyond its working range.
For editorial illustration assignments calling for "Escher-feel" or "ambiguous space," 5-point fisheye is the foundation. Even if the final image crops to a rectangle, the construction starts in the full circle.
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