Exercises: Perspective projection

Explain in your own words what a perspective projection offers compared to the three views of a part: who finds it easier to understand, what can be seen at a glance, and what information, on the other hand, is lost or distorted compared to the views (think about the actual measurements of each face).

In cabinet projection, the front face of the part is drawn in its true shape and size, and only the depth is drawn inclined. Explain what advantage this has when drawing objects with an important front face (a piece of furniture from the front, a building façade) and which face you would choose as the 'front' face when drawing a bookcase in cabinet projection.

You can do it on paper and send a photo with your answer.

The figure shows the same cube drawn in two different perspective projections, with its axes labelled, but without stating which type each one is.

Look at the front face of each cube and its axes, and state which is the cabinet projection and which is the isometric projection, justifying your answer by how the front face is drawn and how the three axes are arranged.

You can do it on paper and send a photo with your answer.

In isometric projection, the three axes form equal angles with each other and no face is seen 'from the front': all faces appear equally inclined. Explain the advantage of treating all three faces equally and why, in isometric drawing, edges that are parallel on the part are also drawn parallel.

A classmate draws a wardrobe in isometric projection and complains: "there is no way to see the door from the front". Explain whether they are right, which perspective projection they should have used if they want to show the door in its true shape, and what is lost when changing from one to the other.

You can do it on paper and send a photo with your answer.

Match each situation with the perspective projection you would choose, justifying your choice: (a) showing the design of a bookcase to the carpenter highlighting its front, (b) drawing a workshop part with similar faces on all three sides, and (c) making a quick sketch of a classroom to arrange furniture, with the side walls visible.

You are given descriptions of two drawings of the same die: in the first, the face showing 6 is seen as a perfect square and the rest of the faces extend distorted towards the back; in the second, the three visible faces are identical rhombuses. State which perspective projection each drawing represents and reason in which of the two you could directly measure the face showing 6 with a ruler.

In a perspective view of stacked cubes, three levels are visible: a row of four cubes at the bottom, a row of two above it, and a single cube at the top. Calculate how many cubes the structure contains if there are no hidden gaps, and explain what precaution must be taken when counting cubes in a perspective drawing (think of cubes that might be hidden behind others).

The figure shows two adjacent cubes drawn in isometric projection: the left one is complete, but the right one is missing its top face, which is covered up.

Explain which four edges must be drawn to complete the top face of the second cube (its shape and inclination), using the clues in the figure: which already drawn edges must be parallel to the new ones.

You are given the three views of a part: the front view is an L-shape made of three squares (two at the bottom and one at the top left), the plan view consists of two squares in a row, and the side view consists of two stacked squares. Reconstruct the part in your notebook in perspective projection and describe the step-by-step process: which view you use for the base, how you place the top cube, and how you check that your perspective view matches all three views.

A part shown in perspective projection is an elongated block with a small cube on top of one of its ends. Describe how to obtain its three views from the perspective drawing: from where you look for each view, what shape each view has, and in which of them the position of the small cube is best appreciated.

To draw in isometric projection on grid paper, you first mark the three axes from a vertex and then measure the edges along them. Write step-by-step instructions to draw a cube with an edge length of $3\ \text{cm}$ in isometric projection, so that a classmate can follow them without help. Include how to close the faces using straight lines parallel to the axes.

A cube has the word STOP written on its front face, in capital letters and well centred. Explain how that face will look if you draw the cube in cabinet projection (with that face as the front) and how it will look if you draw it in isometric projection: in which of the two will the word be read without distortion and why?

These three errors are typical when drawing in isometric projection for the first time: (a) drawing edges that are parallel on the part so that they intersect in the drawing, (b) drawing the top face as a horizontal rectangle, and (c) forgetting the hidden edges where the shape changes. Explain what is wrong in each case and how to correct it, drawing on the properties of isometric projection.

In isometric projection, if the bottom front vertex of a cube is at the point you choose, the edges extend from it in three directions: vertical and two symmetrical oblique directions. Draw or describe how you would locate the vertices of a block measuring $4\ \text{cm}$ long, $2\ \text{cm}$ deep and $2\ \text{cm}$ high, indicating the length of each edge in the drawing and the direction in which each extends.

You can do it on paper and send a photo with your answer.

A staircase with three equal steps goes up from left to right. Describe its isometric projection (what shape the side silhouette has, how many horizontal faces are visible) and explain which view — front view or plan view — best shows the 'step' shape and which shows only rectangles.

The figure shows a stepped piece in isometric perspective along with its front view (elevation) and top view (plan), but the left side view (located on the right according to the European/first-angle projection system) is covered.

A classmate states: "The covered side view must be a plain rectangle 3 units high by 1 unit wide without any interior lines, because from the side you only see a flat vertical wall."

1. Refute their statement: deduce and describe precisely what that side view must look like exactly (indicating its outline and the visible or hidden lines that must be drawn), justifying from which exact side of the perspective you must look to obtain it according to the European system.

2. Explain what principle of correspondence between views allows you to determine, without needing to measure the perspective, the total height and the intermediate divisions that must match between the front view and said side view.

In the Technology workshop, a piece consisting of a wooden prism of $4\ \text{cm} \times 2\ \text{cm} \times 2\ \text{cm}$ pierced through from top to bottom by a centered through-square hole of $1\ \text{cm} \times 1\ \text{cm}$ is going to be manufactured. A student submits for its manufacture only a drawing in isometric perspective showing the upper opening of the hole and states: "There is no need to make more drawings, in 3D the entire piece is perfectly understood."

1. Refute their statement by pointing out two essential manufacturing details of the piece that the perspective leaves completely indeterminate (ambiguous or invisible) for the workshop operator.

2. Determine which minimum orthographic views must be drawn to define the piece without ambiguity, describing exactly in which of them dashed lines (hidden edges) must appear and what these lines represent.

A team of students designs a modular structure using identical cubic blocks. When projecting its standard orthographic views, they obtain:

• A front view with a continuous rectangular silhouette measuring $5$ cubes wide by $3$ high.

• A top view with a continuous rectangular silhouette measuring $5$ cubes wide by $2$ deep.

One of the group members states: "To build this structure we strictly need $30$ cubes."

1. Prove through spatial reasoning why their statement is wrong: calculate what the MINIMUM number of cubes necessary is to obtain exactly that same front view and that same top view, assuming that all cubes must rest on the base or on another cube. Justify how they should be distributed.

2. Propose which standard technical view or which graphic resource of technical drawing would immediately resolve any ambiguity about the actual number of cubes used, justifying your choice.

A furniture workshop designs a display cabinet whose front includes a door with circular moldings and precision handles. To coordinate the project, the novice designer proposes sending a single drawing in isometric perspective to both the private client and the carpenter in charge of machining the parts, arguing that "isometric is the most complete system because it shows realistic volume and allows dimensions to be taken directly with a ruler."

1. Identify and justify two serious technical errors in the designer's proposal regarding the carpenter's work (consider direct measurement of dimensions and the deformation suffered by circular moldings in isometric compared to orthographic views).

2. Justify which specific representation system is appropriate to send to the client and which to the carpenter, explaining in which perspective (cavalier or isometric) it would be advisable to represent the door with circular moldings if you wanted to draw the front without any geometric deformation in its edges and angles.

On the drawing of an auxiliary piece of furniture drawn in cavalier perspective (Y axis at $135^\circ$ with a reduction factor of $0.5$), a workshop apprentice must cut the depth crossbars and drill a circular opening in one of the boards. They make two interpretation errors:

• They measure the depth of the piece of furniture directly on the oblique axis of the drawing with a millimeter ruler, read $15\ \text{cm}$, and cut the slats to $15\ \text{cm}$.

• They draw a perfect circle with a compass directly on the top face of the drawing to represent the cable pass-through hole.

• Correct the first error: calculate the actual length the cut slats should have, formally justifying how the reduction factor is applied between the real piece and its representation on the depth axis.

• Explain what geometric error they made in the second case: describe what shape a circle projected onto the top face in cavalier perspective actually takes and on which single face of this system a circular hole is drawn as a perfect circle with a compass.

A classmate tries to draw a stepped support with a rectangular base in isometric perspective. When reviewing their drawing, you notice that they have drawn the top corners of the steps at $90^\circ$ on the paper "so that it can be seen that the steps are square" and have tilted the vertical edges toward the center "to give a sense of perspective toward the background."

1. Explain why both reasonings are geometrically incompatible with isometric perspective: justify what exact angle the three isometric axes make with each other on the paper and what immutable orientation the vertical edges must always maintain.

2. If the base of the step measures $60\ \text{mm}$ in width, $30\ \text{mm}$ in depth, and $20\ \text{mm}$ in height, describe the exact geometric procedure to draw it correctly using lines parallel to the three isometric axes and explain how you would check with a ruler and set square that no parallel edge has been deformed.

A signage company commissions the manufacturing of a 3D channel letter 'T' made of sheet metal with the following dimensions: total height of $20\ \text{cm}$, width of the upper crossbar of $16\ \text{cm}$, thickness of each arm of $4\ \text{cm}$ in all its parts (crossbar of $16 \times 4\ \text{cm}$ and centered vertical stem of $4 \times 16\ \text{cm}$), with a uniform depth of $5\ \text{cm}$ going backwards.

1. If you are going to draw it in cavalier perspective, technically justify which face you should place parallel to the frontal plane and what the drawn length of the receding edges should be if you apply a reduction factor of $0.5$.

2. Accurately determine the outer dimensions (width $\times$ height) of the three main orthographic views (front view, top view, and left side view) and reasonably explain whether any hidden (dashed) edge should appear in the top view and why.

Maurits Cornelis Escher and other artists created famous lithographs featuring "impossible figures", such as a continuous four-flight staircase that ascends endlessly and reconnects with itself in a closed loop.

1. Analyze the technical basis of this visual trick: what property of projection onto a two-dimensional flat surface makes it possible to deceive human perception by making us believe that lines located at different depth planes meet at the same vertex?

2. If a draftsperson tried to draw the standardized orthographic views (front view, top view, and side view) of that infinite staircase, demonstrate what insurmountable contradiction would immediately appear when trying to maintain the correspondence of heights (elevations) between the front view and the side view.