The mathematics of sidewalk illusions - Fumiko Futamura

譯者: Winston Szeto 審譯者: Helen Chang

If you're ever walking down the street

若你在街上步行時

and come across an oddly stretched out image, like this,

遇見這樣一個形狀怪異的拉闊圖像

you'll have an opportunity to see something remarkable,

假如你站在正確的位置

but only if you stand in exactly the right spot.

就有機會看到精彩的圖像

That happens because these works employ a technique called anamorphosis.

因為這些圖像採用了 「3D 立體畫」的技巧

Anamorphosis is a special case of perspective art,

3D 立體畫是透視投影藝術的特例

where artists represent realistic three-dimensional views

是藝術家在二維平面上

on two-dimensional surfaces.

呈現如假似真的三維視覺效果

Though it's common today,

雖然現在很普遍

this kind of perspective drawing has only been around since the Italian Renaissance.

這種透視投影卻始自 意大利文藝復興時期

Ancient art often showed all figures on the same plane,

古藝術作品通常 把一切圖形擺在同一平面上

varying in size by symbolic importance.

以尺寸大小來象徵它的重要程度

Classical Greek and Roman artists realized they could make objects seem further

古典希臘和羅馬藝術家 明白能以縮小的圖像

by drawing them smaller,

來代表遠處的事物

but many early attempts at perspective were inconsistent or incorrect.

但早期的透視投影 大多不一貫或甚至錯誤

In 15th century Florence,

在十五世紀的佛羅倫斯

artists realized the illusion of perspective

藝術家明白透視投影的錯覺

could be achieved with higher degrees of sophistication

能透過運用數學原理

by applying mathematical principles.

變得更加精密複雜

In 1485, Leonardo da Vinci manipulated the mathematics

達文西在 1485 年利用數學原理

to create the first known anamorphic drawing.

製成有史以來首幅 3D 立體畫

A number of other artists later picked up the technique,

後來有幾位藝術家仿效這種技巧

including Hans Holbein in "The Ambassadors."

包括霍爾拜因的畫作《大使們》

This painting features a distorted shape that forms into a skull

畫上呈現一個形狀扭曲的骷髏頭像

as the viewer approaches from the side.

若觀眾從側邊靠近就看得到

In order to understand how artists achieve that effect,

要理解藝術家怎樣做到 3D 立體畫的效果

we first have to understand how perspective drawings work in general.

必須先理解透視投影畫作的一般原理

Imagine looking out a window.

想像往窗外望

Light bounces off objects and into your eye,

光線從物件表面反射過來

intersecting the window along the way.

穿過窗戶射進眼睛

Now, imagine you could paint the image you see directly onto the window

然後想像把看到的影像直接畫在窗上

while standing still and keeping only one eye open.

同時挺立不動，只張開一隻眼

The result would be nearly indistinguishable from the actual view

結果跟實際影像幾乎一模一樣

with your brain adding depth to the 2-D picture,

那是因為大腦把景深加在二維影像上

but only from that one spot.

立體視覺的效果只限於從特定的點看

Standing even just a bit off to the side

稍微往側邊移動一點點再看

would make the drawing lose its 3-D effect.

立體視覺效果隨之消失

Artists understand that a perspective drawing

藝術家明白透視畫像

is just a projection onto a 2-D plane.

就是在二維平面上的投影

This allows them to use math to come up with basic rules of perspective

他們運用基本的數學觀點規則

that allow them to draw without a window.

不用窗戶也能畫透視圖

One is that parallel lines, like these,

原理之一：像這樣的平行線

can only be drawn as parallel if they're parallel to the plane of the canvas.

跟畫布表面平行才可畫成互相平行

Otherwise, they need to be drawn converging to a common point

否則就要畫成會合在某個共同點

known as the vanishing point.

這一點稱為「消失點」

So that's a standard perspective drawing.

那就是標準的透視投影畫作

With an anamorphic drawing, like "The Ambassadors,"

像《大使們》一類的 3D 立體畫

directly facing the canvas makes the image look stretched and distorted,

正面朝向畫布看到的 是被拉長扭曲的圖像

but put your eye in exactly the right spot way off to the side,

但是把眼睛放在側面恰好的位置

and the skull materializes.

骷髏頭就成形了

Going back to the window analogy,

以剛才的窗上畫圖比喻來解釋

it's as if the artist painted onto a window positioned at an angle

骷髏頭就像從側邊畫上窗戶一樣

而非正對畫在窗戶上

instead of straight on,

though that's not how Renaissance artists actually created anamorphic drawings.

雖然那不是文藝復興藝術家 畫 3D 立體畫的方法

Typically, they draw a normal image onto one surface,

他們通常把正常圖像畫在平面上

then use a light,

然後用光照射

a grid,

用網格

or even strings to project it onto a canvas at an angle.

甚至用弦線把圖像用個角度 投影在畫布上

Now let's say you want to make an anamorphic sidewalk drawing.

假設你要畫一幅 人行道上的 3D 立體畫

In this case, you want to create the illusion

要引起別人的錯覺

that a 3-D image has been added seamlessly into an existing scene.

以為一幅立體影像無縫地 嵌在現有的間隙中

You can first put a window in front of the sidewalk

方法是先拿着一扇窗面對人行道

and draw what you want to add onto the window.

把要畫在人行道的圖先畫在窗上

It should be in the same perspective as the rest of the scene,

它應該和其他場景一樣

which might require the use of those basic rules of perspective.

可能需要使用這些基本的視角規則

Once the drawing's complete,

圖畫好以後

you can use a projector placed where your eye was

把投影機置於眼睛的位置

to project your drawing down onto the sidewalk,

將圖投影到人行道上

then chalk over it.

然後用粉筆填色

The sidewalk drawing and the drawing on the window

人行道上圖像和窗上的圖像

will be nearly indistinguishable from that point of view,

從那個視點來看應該幾乎一樣

so viewers' brains will again be tricked

這使看畫的人產生錯覺

into believing that the drawing on the ground is three-dimensional.

以為地面上的圖像是立體的

And you don't have to project onto a flat surface to create this illusion.

這種錯覺未必得投在平坦的表面上

You can project onto multiple surfaces,

也可以投影在多個平面上

or assemble a jumble of objects, that from the right point of view,

也可透過拼湊無數物體來形成錯覺

appears to be something else entirely.

從合適視點可看到完全不同的東西

All over the planet, you can find solid surfaces

世界上有許多的地面

giving way to strange, wonderful, or terrifying visions.

被畫上各式奇形怪狀 栩栩如生或駭人的圖像

From your sidewalk to your computer screen,

無論是在人行道或者電腦上觀看

these are just some of the ways that math and perspective

這些 3D 立體畫作 都是數學原理和透視法結合後

can open up whole new worlds.

創造無限可能的例子