Anamorphic Street Art Projection Calculator - Perspective Distortion Planner

The Magic of Anamorphic Illusions

Anamorphic street art transforms sidewalks and plazas into bewildering scenes that pop into place from a single viewing spot. The artist paints a wildly stretched design on the pavement which, when seen from a carefully chosen location, appears as a normal upright object. Popular examples include gaping chasms, floating bridges, or towering monsters that startle passersby. The trick relies on perspective projection: lines that are parallel in the real world converge toward vanishing points on the picture plane. By inverting this projection, we can determine how to distort an image on the ground so that the observer’s eye reconstructs the intended scene. The calculator here automates this geometric reasoning for a simple case — making a rectangle on the ground appear as a vertical billboard. It accepts the apparent width and height you want, the height of the viewer’s eyes above the ground, and how far in front of the viewer the near edge of the drawing begins. From these it outputs coordinates for the four corners of the distorted quadrilateral you must paint. The math extends to more complex shapes, but mastering the rectangle builds intuition.

Setting Up the Geometry

We adopt a right-handed coordinate system with the ground as the $x - y$ plane and the $z$ axis pointing upward. The viewer stands at the origin with eyes at height $h$ along $z$ . They look down the positive $y$ axis. We want a planar rectangle to appear standing perpendicular to the ground at some distance. Conceptually, imagine a vertical screen located at $y = L$ . On that screen the rectangle spans width $W$ along $x$ from $- \frac{W}{2}$ to $\frac{W}{2}$ and height $H$ along $z$ from 0 to $H$ . To make the drawing appear correct, we imagine rays emanating from the viewer’s eye through the corners of this ideal rectangle. Where each ray intersects the ground plane gives a point on the pavement to paint.

Projecting the Corners

The corner coordinates of the desired upright rectangle in the screen plane are:

Corner	$x$	$y$	$z$
Bottom Left	$- \frac{W}{2}$	$L$	0
Bottom Right	$\frac{W}{2}$	$L$	0
Top Left	$- \frac{W}{2}$	$L$	$H$
Top Right	$\frac{W}{2}$	$L$	$H$

For any point $(x, L, z)$ , a line from the eye position $(0, 0, h)$ to that point can be parameterized as $(x t, L t, h + (z - h) t)$ . The intersection with the ground plane $z = 0$ occurs when $t = \frac{h}{h - z}$ . Applying this to the top corners where $z = H$ yields a magnification factor $t = \frac{h}{h - H}$ . We denote this factor by $\tau$ , representing how much farther the top corners must be from the eye compared to the bottom corners. The ground coordinates of the distorted quadrilateral become:

$(- \frac{W}{2} \tau,\, L\tau,\,0)$ and $(\frac{W}{2} \tau,\, L\tau,\,0)$ for the top corners, while the bottom corners remain at $(- \frac{W}{2},\, L,\,0)$ and $(\frac{W}{2},\, L,\,0)$ .

Area and Distortion

The painted shape is a symmetric trapezoid whose top edge is $W\tau$ wide and whose bottom edge is $W$ . Its depth on the ground is $L(\tau - 1)$ . The area follows the trapezoid formula $A_p = L(\tau - 1)$ . We compare this to the actual area of the intended upright rectangle $A_r = W H$ . The distortion ratio $\eta = \frac{A_p}{A_r}$ indicates how much extra pavement the artist must cover. When $H$ approaches the viewer’s eye height $h$ , $\tau$ grows large, the trapezoid stretches dramatically, and $A_p$ balloons. This is why photos of anamorphic art from the wrong angle look absurdly elongated.

Visualizing the Output

Imagine you desire a 2 m by 1.5 m sign. Your eyes are 1.7 m above the ground and you stand 1 m from the near edge. The magnification factor becomes $\frac{1.7}{1.7 - 1.5} = 8.5$ . The top edge must therefore be eight and a half times farther and wider than the bottom edge: over 17 m away with a span of 17 m. The resulting trapezoid covers more than 76 m², around twenty-five times the area of the perceived rectangle. The calculator performs this arithmetic instantly and delivers the four corner coordinates ready for chalking.

Practical Considerations

Anamorphic drawings rely on guiding the audience to the correct vantage point. Chalk artists often mark footprints or place a camera tripod at the ideal spot. If viewers roam, the illusion collapses. Because our model assumes the viewer’s eye height exceeds the desired apparent height, the artist may need to mount a camera low to the ground and design the illusion for that specific lens. Alternatively, crouching viewers can match the requirement. Lighting and surface texture also affect perception; glossy pavement can create specular highlights that break the illusion, while rough textures diffuse lines. If the ground is sloped, you must adjust the geometry accordingly. Our calculator presumes a flat plane.

Extending the Method

While we present only a rectangle, the technique generalizes. For arbitrary shapes, divide the desired image into small polygons or a grid. Project each vertex using the same formula for $\tau$ . Artists sometimes use software to warp an image photographically: a digital camera at the viewer position captures the blank pavement, and the desired artwork is inverse-projected onto that photo. The printed result serves as a guide. Our calculator offers a transparent, mathematical approach that aids understanding and can be executed with simple tools.

Typical Values

The table below shows distortion ratios for several viewer heights and perceived heights with a 1 m near edge and 2 m width:

$h$ (m)	$H$ (m)	$\tau$	$\eta$
1.8	1.0	1.8	2.3
1.8	1.4	4.5	9.0
1.8	1.6	9.0	25.5

As the desired height approaches the eye height, the distortion ratio soars, underscoring how demanding large illusions can be. Artists mitigate this by selecting viewing positions above the artwork, such as stairs or balconies, which increase $h$ and reduce $\tau$ .

Using the Calculator

Enter the apparent width and height of your intended figure, your eye height, and how far the chalk starts from your feet. Press “Project Rectangle” to receive ground coordinates: the near left and right points, followed by the far left and right points. You can copy the text to the clipboard. Transferring the coordinates to the pavement is straightforward: align a tape measure along the centerline for the $y$ direction, mark the specified distances, then measure perpendicular offsets for the $x$ coordinates. Connecting these four points with straight lines yields the anamorphic trapezoid. Fill in your artwork within this shape, referencing the stretched proportions.

Limitations and Creativity

The algorithm assumes a single viewer. If people approach from different positions, perspective cues conflict and the illusion breaks. Real scenes are rarely perfectly rectangular; curved or irregular shapes require more vertex projections. Atmospheric effects such as heat ripples or shadows can distract the eye. Nevertheless, embracing these challenges leads to captivating art. Some creators incorporate real objects into the drawing, aligning them with painted portions to further deceive viewers. Others design illusions that appear correct only through a smartphone camera, letting the device stand in as the fixed eye position. With practice, you can expand the calculator’s principles to craft entire 3D scenes.

Conclusion

Anamorphic street art offers a playful fusion of geometry and imagination. By quantifying the distortion needed for a simple rectangle, this calculator demystifies the underlying projective geometry and empowers artists to plot their own illusions. Whether chalking a sidewalk for a festival or planning a public installation, understanding how eye height and distance influence the trapezoid on the ground ensures your artwork snaps into place for astonished viewers. Experiment with different parameters to appreciate the dramatic stretch required, then grab your chalk and transform an ordinary pavement into a portal to another dimension.