In 2D, a true version of what is seen is not created, only emulated - mainly because the world is arguably 3D.

There is information loss associated with the transformation from reality to a 2D medium emulation.

Creativity comes into play in how this information loss is handled.

2D emulations of a 3D scene: photos from cameras, sketches on paper, and digital renders of 3D graphics - these are all an approximation of how the 3D world is seen - an illusion that mimics the 3D world

Human vision is stereoscopic, but 2D emulations are generated from the perspective of one eye

In photography, this is the camera sensor

In digital packages like Blender and Houdini, a virtual camera is used.

In paper sketches, it is the best estimate made by the draw-er.

Rules of Illusion

Perspective principles are the fundamental rules of creating a 2D emulation of the 3D world. They prescribe paradigms for methodically graphing a 3D scene on to a 2D plane.

There are few occasions that call for the breaking of these rules - commentary on the rules being one of them.

Else, breaking perspective principles makes the illusion tacky, and/or distracts from the goal. These distractions might not necessarily be things that the viewer may be able to pin-point to, but still take away from the point of the piece.

Perspective concepts are many and deep, and requires considerable research to be fully understood. However, there are foundational concepts that the rest of the conversation is built on.

In digital systems, perspective rules are usually automated in producing 2D emulations i.e. photos and renders.

In paper sketching, the draw-er needs to be aware of perspective paradigms while generating their work. Those paradigms are explored here.

Goal

The goal of perspective drawing is to be able to find any point that belongs to the 3D space in the resulting 2D space of the drawing.

Mathematically, this is similar to the process of finding the projection of (dot product) vectors of the 3D space on to the 2D plane, i.e. finding the projection of a space over a sub-space.

In paper 2D sketches, the draw-er manually finds all the projections of the 3D scene points graphically using lines and curves.

In digital graphic packages, the light paths the virtual camera sees are calculated numerically using vector algebra and integration.

In photography, the camera sensor simply captures all the light received at the pixel site.

Scene

The scene is the visible region of the 3D volume in consideration. It is very dependent on the point-of-view of observation. It may be real or fictional.

In photography, this is bound by the camera sensor and lens combination. Similarly so in 3D graphics packages.

In paper sketches, this is usually chosen by the draw-er before commencing the drawing.

Coordinate System

  • 3D space is defined by three mutually perpendicular axes:
    • X-axis,
    • Y-axis and
    • Z-axis
  • each axis points towards a Vanishing Point (VP)
  • this results in three planes:
    • X-plane
    • Y-plane
    • Z-plane
  • each axis is perpendicular to its plane
  • all 3D scenes are subsets of this coordinate system

Vanishing Point (VP)

  • these are points that lines go to at infinity in the scene
  • depending on the perspective, there may be
    • Center Vanishing Point (CVP)
    • Left Vanishing Point (LVP)
    • Right Vanishing Point (RVP)
    • Bottom Vanishing Point (BVP)
    • Top Vanishing Point (TVP)

Horizon Line (HL)

  • the lines in a plane extended to infinity into the depth of the scene converge at the Horizon Line
  • the HL has an infinite number of VPs

3D-perspective-space

fig: X-axis (red), Y-axis (green), Z-axis (blue), horizon line (orange)

Defining a Perspective

  • to control perspective, it is necessary to clearly define the following
    • where the viewer is standing
    • the viewing direction
    • the vision’s lens angle
  • this applies to
    • guessed perspective
    • constructed perspectives
    • computer-generated perspectives

Ground Plane (GP)

  • the flat plane on which different objects in the scene rest
  • the plane the photographer stands on while shooting a picture
  • the height axis is perpendicular to the ground plane
    • usually, the Z-plane

Station-Point (SP)

  • point in space above the GP where the eye or the camera sensor is located

Line of Sight (LoS)

  • direction and incline of the observing eye

Picture Plane (PP)

  • the plane on which the 2D emulation of the 3D scene is formed
    • the final render or the drawing is what results on the PP
  • the final paper sketch will have the light rays from the 3D scene intersecting this plane
  • in a camera, this is the plane of the sensor
  • perpendicular to the LoS
  • the farther the PP is from the SP, the larger it is
    • so the drawing/rendering is bigger for the same scene
    • but the proportions of the elements in the scene are still the same

Distortion

  • looking in the 3D space graphic below,
    • GP grid squares are minimally distorted away from being squares somewhere in the center of the image height
      • the squares close to the PP are distorted by expansion
      • the squares further away from the sweet central region are distorted by compression
  • PP and SP are generally set to put the subject in least distortion
    • in some instances, the distortion is deliberately used artistically

fig: SP - station point, HL - Horizon Line, PP - picture plane in 3D space, GP - grid floor

Cone of Vision (COV)

Lenses

  • right off the bat, it helps to use the lens paradigm from photography to understand COV
  • COV of lenses are dependent on their focal length and the size of the image sensor they are paired with
  • lenses range from wide-angle to telephoto; for full-frame (35 mm) camera sensors
    • 8 - 11 mm lenses are wide-angle
    • 35 mm lenses are standard
    • 50 mm - 105 mm are portrait
    • 200 mm - 300 mm lenses are telephoto
  • 35 - 50 mm lenses provide largely distortion free images
  • 35 mm lens corresponds to approximately 60º COV
    • 60º COV is the go-to for most drawings

wide-angle

  • as the focal length reduces toward 8 mm, the squares closer to the picture plane get bigger and add to distortion
    • wide angles lenses are great for capturing large scenes
  • these lenses have COV larger than 60º
  • these lenses capture the depth of the scene better relative to their longer focal length peers when the subject is spread out
  • anything below 8 mm is fish-eye,
    • distorts the image by copious amounts
    • this is sometimes used artistically to add dramatic effect

telephoto

  • as the focal lengths get longer, closer to 300 mm and over it, the captured image is of a small piece of the whole scene
    • but the details of this small region is captured well
    • often called zoom lenses also
  • 300 mm crop-sensor lens has ~ 5º COV
  • the image, however, has a “flattened” look to it because the squares are evenly sized in the depth of the captured scene
  • good for wildlife, action sports, and planetary imaging type subjects

COV Spectrum


fig: lens focal length spectrum


fig: 24 mm, 35 mm and 100 mm lenses at the SP


fig: 24 mm COV in above setup


fig: 35 mm COV in above setup


fig: 100 mm COV in above setup

Perspective Boxes

When making a 2D emulation, the idea is to fully fill the space in the medium bounds.

Using more VPs add more information in the available space. However, in the case of manual sketching, they make the process more complex.

2D emulations are classified by the number of VPs that used. A boundary box is constructed using the VPs used. Then the subject is drawn into it using perspective grid techniques.

1-Point Linear Perspective Box

  • only 1 VP in the COV grid, so prone to distortions
  • to minimize distortions use 50º COV
    • 40º COV is even better to avoid distortions
  • however, smaller the COV, flatter the perspective box becomes
    • depth information degrades

2-Point Linear Perspective Box

  • the 1 VP COV is extended by using a second VP to reduce distortion
  • distortion increases towards the edge
    • avoid placing critical drawing elements at the edge
  • 60º COV is the go-to COV

3-Point Linear Perspective Box

  • a third VP is added along a line perpendicular to the line connecting the first two VPs
    • the COV is further extended
  • edges are distorted similar to the 2 - PT box
  • stick to 60º COV

COV sphere

  • COV extended out in all directions generates a sphere
  • considering a 60º COV,
    • 1-VP perspective box is inside the COV
    • 2-VP box is outside the COV, borrowing one VP to reduce distortion
    • 3-VP box borrows two VPs outside the COV to accommodate more

fig: COV sphere with 1-PT, 2-PT and 3-PT perspective boxes

5-Point Curvilinear Perspective Box

  • this box ceases to be made of straight lines
    • similar to fish-eye lenses shots
  • this perspective allows more that natural field-of-view perception on 2D mediums
  • managing the projection lines is a huge challenge
    • as intuition doesn’t work in the draw-er’s favor
  • 3-point boxes are already challenging enough with straight lines
    • even with using a straight-edge for construction

Line Convergence

  • as a general rule, physical parallel lines converge to a vanishing point
    • there are exceptions in the 1-PT and 2-PT perspectives
  • lines that converge to a VP are harder to construct as more points are needed to be identified accurately to be able to draw the line
  • lines that do not converge are typically parallel to GP or PP, and their construction is simpler

1-PT perspective lines

  • only converge into the depth of the drawing
  • lines parallel to PP i.e. perpendicular to the line of sight do not converge
  • consequently, the 1-PT box setup is faster, with only one VP

2-PT perspective lines

  • all physical parallel lines converge, except the verticals
  • since verticals are perpendicular to HL, it makes these construction lines easy to draw
  • outside the 60º COV, the drawing get distorted

SP - HL relationship

LoS, HL and COV move in tandem. The visible information depends on the COV sphere.

LoS parallel to GP

  • as the SP moves up or down, the HL moves proportionally
  • this also affects how much of the object being observed remains in the COV
  • verticals are perpendicular to the HL in 2-PT perspective

LoS not parallel to GP

  • verticals begin to converge
  • when looking up, base of the object is hidden out of sight
  • when looking down, top of the object is hidden out of sight

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