Three-Dimensional Figures and Graphs

Transcription

1 UNIT 6 Three-Dimensional Figures and Graphs We live in a three-dimensional world that is often represented on a two-dimensional plane. Drawings, paintings, and maps are some of the two-dimensional representations of our world. Artists were not alwas able to accuratel represent three dimensions on a flat surface. In the fifteenth centur, Filippo Brunelleschi, an Italian sculptor and architect with an interest in mathematics, was the first to conduct eperiments that led to an understanding of one-point perspective. To create the illusion of depth on a twodimensional surface, he demonstrated that lines that are parallel in the real world should be drawn as meeting at a single vanishing point. In this unit, ou will etend our geometric knowledge and skills to three-dimensional figures. UNIT OBJECTIVES draw basic three-dimensional figures as well as twodimensional views of three-dimensional figures. define polhedron and identif the faces, edges, and vertices of polhedra. define prism and the parts of a prism. Classif prisms. find the surface area of prisms. find the volume of prisms. locate and plot points in a three-dimensional coordinate sstem. Use the Distance and Midpoint Formulas for three dimensions. Use intercepts to graph planes in space. Use parametric equations to plot lines in space. Unit 6 three-dimensional figures and graphs 173

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3 Solid Shapes and Three-Dimensional Drawing Objectives Define three-dimensional drawing and depth. During the Renaissance, artists started using perspective to represent the three-dimensional world on their two-dimensional canvases. Notice that the painting on the right has depth and looks more realistic than the one on the left. Use isometric grid paper to show how threedimensional figures can be drawn. Draw two-dimensional views of three-dimensional figures. Interpret a threedimensional drawing. Solve problems involving surface area and volume. KEYWORDS depth orthographic views solids three-dimensional drawing isometric drawing perspective drawing surface area volume Figures in Three Dimensions Polgons are two-dimensional figures. The are flat and lie on a single plane. Solids are three-dimensional figures. Three-dimensional figures have the additional dimension of depth. Depth is a measure of the length of a figure from the front of the figure to the back. Solid Shapes and three-dimensional drawing 175

4 vanishing point Drawing Three-Dimensional Figures A three-dimensional drawing represents a three-dimensional figure on a two-dimensional plane. The three-dimensional drawings shown below are called perspective drawings because the objects appear true-to-life, as the would if ou were actuall looking at them. horizon 1-point perspective -point perspective You can draw a cube in one-point perspective b first drawing a square, a line called the horizon, and a point on the horizon called the vanishing point. This square is the front of the cube. Draw straight lines from the vertices of the square to the vanishing point. Draw the back face of the cube so that its vertices are on the lines leading to the vanishing point. You ma find it easiest to draw the top horizontal segment first. Because it is a cube, locate the points so that all sides appear to be the same length. Connect the front and back squares b tracing over the segments between them. To draw a cube in two-point perspective, start with the horizon, two vanishing points, and a vertical line segment that will be the front edge of the cube. Draw lines from the front segment s endpoints to each vanishing point and use those to draw the left and right vertical sides of the cube. Join the top of each edge to its opposite vanishing point to form the top face of the cube. Look carefull at the sides and angles of each cube in the diagrams. Some sides that should be parallel are not drawn parallel, and some angles do not actuall measure 90. Isometric Drawings of Three-Dimensional Figures Another tpe of three-dimensional drawing is called an isometric drawing. Isometric drawings show three sides of a solid figure from a corner view. Notice in the diagram at left that the parallel lines are drawn parallel and the right angles on the top face are drawn as 10 and 60 angles. This is one wa isometric drawings differ from perspective drawings. You can use isometric grid paper to help ou draw three-dimensional figures. 176 Unit 6 three-dimensional figures and graphs

5 Orthographic Views Orthographic views of a three-dimensional figure show the front, back, top, bottom, left side, and right side views of the figure. The orthographic views of a square pramid are shown in the diagram. 1-point perspective top bottom left side right side front back set of orthographic views To draw a top view, imagine fling above the object and looking straight down on it. If ou flew above the pramid and looked down, ou would see more than just the top point. You would see the outline of the base of the figure as well as the four edges that lead to the corners of the base. For a bottom view, imagine ling below the object and looking straight up. For side views, imagine floating around the figure and recording what ou see when ou face each side directl. The orthographic views of a previous drawing are shown below. Notice that in the left side, right side, front, and back views, the depth of each block is irrelevant. If ou can see a block from that side, regardless of its depth, a square is included in the orthographic view. Just a couple of views of a figure do not tell ou what the figure looks like. Though three views are sufficient for some figures, other figures require as man as si views to get an accurate view of what the figure looks like. isometric drawing top bottom left side right side front back set of orthographic views Solid Shapes and three-dimensional drawing 177

6 1 in. 5 in. 3 in. Surface Area and Volume The surface area of a figure is the amount of space that covers the figure. To find the surface area of a solid, find the sum of the areas of its outer surfaces. Compare the area calculations below with the three-dimensional figure shown at left. top 1 5 = 60 bottom 1 5 = 60 left side 3 5 = 15 right side 3 5 = 15 front 1 3 = 36 + back 1 3 = 36 RECONNECT TO THE BIG IDEA Remember Measurement is the process of using a unit to determine how much of something ou have. 4 cm The surface area of the three-dimensional figure is in. The volume of a solid is the amount of space inside the figure, measured in cubic units. If it s a small rectangular figure, ou ma be able to count the cubes. This is normall not possible, however, because of either the size or the shape of the figure. In a three-dimensional figure like the prism at left, ou can multipl or count to find the area of one laer of cubes and multipl b the number of laers. Because each laer in this figure is a rectangle, ou can find its volume b multipling its dimensions. V = = 60 3 cm 5 cm The volume of the figure is 60 cm 3. Summar Perspective drawings show three-dimensional objects as the appear in real life. To draw in perspective, use a horizon and vanishing points. Isometric drawings show three sides of a solid figure from a corner view. Orthographic views of a three-dimensional figure show the front, back, top, bottom, left side, and right side views of the figure. The surface area of a solid is the amount of space that covers the figure. The volume of a solid is the amount of space inside the figure. 178 Unit 6 three-dimensional figures and graphs

7 Lines, Planes, and Polhedra Objectives Identif and define skew lines, half-planes, and dihedral angles. Define polhedron. A globe is a fair representation of the earth it is a three-dimensional model representing a three-dimensional shape. However, flat maps of the earth, called map projections, distort the earth s shape. Projecting a curved figure onto a flat surface alwas creates some tpe of deformation. Unlike the shape of the earth, man other three-dimensional shapes can be created accuratel from two-dimensional models. Identif the faces, edges, and vertices of polhedra. Create a figure from its net, and draw nets of polhedra. KEYWORDS dihedral angle face net polhedron verte of a polhedron edge half-plane parallel planes skew lines Lines and Planes in Space If two lines lie on the same plane, then those lines either intersect or are parallel. If two lines lie in different planes, there is another possibilit. The lines can be skew. Skew lines are two lines that do not intersect but are not parallel. The blue lines in the diagram are skew lines. Parallel planes are planes that do not intersect. For instance, the top and bottom faces of a cube lie on parallel planes. A line that lies in a plane separates the plane into two half-planes. A halfplane consists of all the points on either side of the line. half-plane lines, planes, and polhedra 179

8 Remember An angle is two ras joined at a point. A dihedral angle is two half-planes joined at a line. If two planes intersect, the form four angles and four half-planes. Each angle is called a dihedral angle. A dihedral angle is formed b two noncoplanar half-planes and their line of intersection. dihedral angles Polhedra A solid is a three-dimensional figure. A polhedron is a solid enclosed b polgons. So, although a sphere is a solid, it is not a polhedron. Note that the two plurals of polhedron are polhedra and polhedrons. The flat surfaces of a polhedron are called the faces. The faces meet to form the edges of the polhedron. The vertices of the faces meet to form the vertices of the polhedron. The number of faces, edges, and vertices of three polhedra are shown below. rectangular prism 6 faces 1 edges 8 vertices square pramid 5 faces 8 edges 5 vertices octahedron 8 faces 1 edges 6 vertices Nets of Polhedra A net is what a solid looks like if ou unfold it. You can think of a net as a pattern for making a solid, much in the same wa that a pattern is used to sew pieces of clothing together. Nets are often used to find surface area. net of a square pramid 180 Unit 6 three-dimensional figures and graphs

9 To create the net of an octahedron with geometr software, create an equilateral triangle, and then use rotations and reflections to create the net in the diagram. If ou like, ou can add color. Once ou have created our net, ou can print it and fold it to create the polhedron. You can use grid paper to create a net. To make sure the figure folds up properl, be sure the top and bottom faces are congruent and that the height for the front and back is the same as for the top and bottom. Summar Skew lines do not lie in the same plane and do not intersect. Parallel planes are planes that do not intersect. A line in a plane separates the plane into two half-planes. An angle formed b the intersection of two planes is called a dihedral angle. A polhedron is a solid enclosed b polgons. Each polgon is called a face. The faces intersect at edges. The vertices of the faces meet to form the vertices of the polhedron. A net shows what a solid looks like when the solid has been unfolded. A net can also serve as a pattern for a solid. lines, planes, and polhedra 181

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11 Prisms Objectives Define prism and the parts of a prism. In the picture, a triangular prism is dispersing light into all of its different colors. Prisms that reflect, refract, and disperse light are called optical prisms. Optical prisms come in man shapes and sizes. You will learn about various kinds of prisms in geometr and how to measure them. Classif prisms. Solve problems that involve the diagonal of a right prism. Find the surface area and volume of prisms. KEYWORDS base area of a prism lateral edge oblique prism right prism diagonal of a polhedron lateral face prism Tpes of Prisms A prism is a polhedron with two parallel, congruent faces called bases. We call the other faces lateral faces. Notice that lateral faces are parallelograms formed b parallel segments that connect corresponding vertices of the bases. These parallel segments are called the lateral edges. Prisms are classified b the shapes of their bases. If the bases of a prism are triangles, we call it a triangular prism; if the bases are heagons, we call it a heagonal prism, and so on. lateral face bases heagonal prism Prisms 183

12 A prism whose lateral edges are perpendicular to the bases is a right prism. right rectangular prism If a prism is not a right prism, it is an oblique prism. The lateral faces of an oblique prism are not perpendicular to the bases. oblique triangular prism Diagonals A diagonal of a polhedron is a line segment that joins two vertices that are in different faces. Like diagonals of polgons, the do not overlap the sides of the figure itself. A diagonal of a right rectangular prism is shown in red at left. Formula for the Diagonal of a Right Rectangular Prism The length of a diagonal d of a right rectangular prism is d = l + w + h 4 cm 8 cm 10 cm For the right rectangular prism at left: d = = = The length of the diagonal is approimatel 13.4 cm. Surface Area of Prisms Recall that surface area tells how much space covers a figure. The surface area of a prism is the sum of the areas of the bases and the lateral faces. The area of one base is defined as B. The sum of the areas of the lateral faces is defined as L. Because the bases are congruent, we write the sum of the bases as B. 184 Unit 6 three-dimensional figures and graphs

13 Formula for the Surface Area of a Prism The surface area S of a prism, where B is the area of one base and L is the sum of the areas of the lateral faces, is S = B + L Follow the steps to find the surface area of the right triangular prism below. 4 in. 5 in. 3 in. 10 in. Step 1: Find B and B b using the formula for the area of a triangle. 1 A = bh = 1 (3)(4) = 6 The area of one base is 6 in. So B is (6 in ) = 1 in. Remember The lateral edges are all congruent. Step : Find L. There are three lateral faces. A = bh A = bh A = bh = 10(3) = 10(4) = 10(5) = 30 = 40 = 50 L = = 10 Step 3: Find S. S = B + L = = 13 The surface area of the prism is 13 in. Volume of Prisms Recall that volume is the amount of space inside a figure. The volume of a prism can be found b multipling the area of a base B b its height h. The height of a prism is the perpendicular distance between the bases. When the prism is oblique, the height will be different from the length of a lateral edge. Prisms 185

14 Formula for the Volume of a Prism The volume V of a prism, where B is the area of one base and h is the height of the prism, is V = Bh To find the volume of the same right triangular prism for which ou just found the surface area, see the following steps. Step 1: Find B. 4 in. 3 in. 5 in. 10 in. A = 1 bh = 1 (3)(4) = 6 The area of a base is 6 in. Step : Identif h. The height is the distance between the bases. In this figure it is 10 inches. Step 3: Find V. V = Bh = 6(10) = 60 The volume of the prism is 60 in 3. Summar Remember Area is measured in square units. Volume is measured in cubic units. A prism is a polhedron with two parallel, congruent bases and lateral faces that are parallelograms formed b the parallel segments that connect corresponding vertices of the bases. A prism whose lateral edges are perpendicular to the bases is a right prism. A prism whose lateral edges are not perpendicular to the bases is an oblique prism. The diagonal of a polhedron is a line segment that joins two vertices that are in different faces. The length of a diagonal d of a right rectangular prism can be found b using the formula d = l + w + h. The surface area S of a prism, where B is the area of one base and L is the sum of the areas of the lateral faces, is S = B + L. The volume V of a prism, where B is the area of one base and h is the height of the prism, is V = Bh. 186 Unit 6 three-dimensional figures and graphs

15 Coordinates in Three Dimensions Objectives Identif characteristics of a three-dimensional coordinate sstem. Locate and plot points in a three-dimensional coordinate sstem. Use the Distance and Midpoint Formulas for three dimensions. Architects, interior designers, and radiologists are just a few of the professionals that have benefited from three-dimensional computer imaging technolog. Architects use computer software to create three-dimensional computer models of buildings. Interior designers create virtual rooms where a client can change wall color or floor tiling with the click of a mouse. Radiologists use computer imaging to see inside the human bod, using a three-dimensional view, and are able to detect medical conditions at a much earlier stage. This lesson introduces ou to a three-dimensional coordinate sstem. KEYWORDS coordinate plane octant right-handed sstem first octant ordered triple three-dimensional coordinate sstem The Three-Dimensional Coordinate Sstem (3, 6) The coordinate plane ou are familiar with is formed b the intersection of an -ais and a -ais. Each point in the coordinate plane is represented b an ordered pair of real numbers (, ). You can locate and plot a point b first starting at the origin, then moving left or right according to the -coordinate, and finall moving up or down according to the -coordinate. Remember that the positive -direction points to the right, and the positive -direction points upward. Plotting points in three dimensions can be accomplished using a similar method. First, we must add a third ais to the sstem, the z-ais, which will correspond to the third dimension. To visualize the addition of this ais, imagine the ordinar -plane ling flat on a table, with the positive -direction pointing toward ou and the positive -direction pointing to the right, as shown in the figure. Coordinates in Three Dimensions 187

16 Now imagine a third ais, the z-ais, which is perpendicular to both the - and -aes. The positive direction for this new ais will point upward. B adding this ais, we have formed a three-dimensional coordinate sstem. z (0, 0, 0) z Notice that in forming this coordinate sstem, we could have chosen the positive z-direction to point downward. B instead choosing it to point upward, we have defined a right-handed sstem. This name comes from the so-called right-hand rule : If ou imagine wrapping our right hand around the z-ais, with our fingers curling from the positive -direction ais toward the positive -direction, as shown in the figure, then our thumb will point in the positive z-direction. Had we chosen the positive z-direction to point downward, we would have formed a left-handed sstem. In most applications, however, right-handed sstems are used. In a three-dimensional coordinate sstem, the intersection of the -, -, and z-aes creates three intersecting planes, the -, z- and z-planes. These planes divide the sstem into eight parts. Each part is called an octant. In the first octant the values of,, and z are all positive z Each point in a three-dimensional coordinate sstem is associated with a unique ordered triple of real numbers having the form (,, z). Given an ordered triple, ou can locate and plot the corresponding point using the following method. First, start at the origin. Then move forward or backward according to the -coordinate. Net, move left or right according to the -coordinate. Finall, move up or down according to the z-coordinate. 188 Unit 6 three-dimensional figures and graphs

17 For eample, to plot the point (3, 4, 4), begin at the origin, and then move 3 units in a positive direction (forward) along the -ais. Then move 4 units in a positive direction (right) along the -ais. To help show perspective, ou can draw a rectangle in the -plane having one corner at the origin and another corner at (3, 4, 0). Finall, after drawing the rectangle, move 4 units in a positive direction (up) along the z-ais. z z z (3, 4, 4) (0, 0, 0) (0, 0, 0) (0, 0, 0) 4 units 3 units 3 units 3 units 4 units 4 units Finding the Distance Between Points Finding the distance between two points in space is similar to finding the distance between two points on a plane. Distance Formula for Three Dimensions The distance d between the points ( 1, 1, z 1 ) and (,, z ) in space is d = ( 1 ) + ( 1 ) + (z z 1 ) The following is how ou would find the distance between (4, 4, 3) and (5, 0, 6): d = ( 1 ) + ( 1 ) + (z z 1 ) = (5 4) + (0 4) + (6 3) = (1) + ( 4) + (3) = = Coordinates in Three Dimensions 189

18 Finding the Midpoint of a Line Segment The formula for the midpoint of a line segment in space is an etension of the formula for the midpoint of a line segment in the coordinate plane. Midpoint Formula in Three Dimensions The midpoint of a segment with endpoints ( 1, 1, z 1 ) and (,, z ) in space has these coordinates: ( + 1, 1 + z, 1 + z ) The following is how ou would find the midpoint of a line segment with endpoints (8, 5, ) and (4, 6, 0): Summar ( + 1 ( 8 + 4, 1 + z, 1 + z ), 5 + 6, + 0 ) 1 ( 6, 5, 1 ) A coordinate sstem has three aes an -ais, a -ais, and a z-ais. Points in space are named b the use of ordered triples. The aes form an -plane, a z-plane, and an z-plane, which divide the coordinate sstem into octants. To find the distance between two points in space, ( 1, 1, z 1 ) and (,, z ), use the Distance Formula for Three Dimensions: d = ( 1 ) + ( 1 ) + (z z 1 ). The coordinates of the midpoint of a line segment in space with endpoints ( 1, 1, z 1 ) and (,, z ), are ( + 1, 1 + z, 1 + z ). 190 Unit 6 three-dimensional figures and graphs

19 Equations of Lines and Planes in Space Objectives Use intercepts to graph a plane in space. Define trace and find the equations of traces given the equation of a plane. Use parametric equations to plot lines in space. When Ldia went hiking, she took her Global Positioning Sstem (GPS) device and marked the parking lot as her first wapoint. After a few hours of hiking up steep hills and across streams, she realized that she didn t know the wa back. She looked at her GPS device, chose the first wapoint, and selected Go to. Eas! The onl problem came when the route took her off the trail and straight into some poison iv. The GPS device determined a line between Ldia, who was standing on a steep hill, and the parking lot. If she could imagine the route from her point on the hill down to the parking lot, she would be imagining a line in space. Define the equation of a plane in space. KEYWORDS intercept parametric equation trace Equations in Space On the coordinate plane, an intercept is where a line crosses an ais. In space, an intercept is a point where a plane crosses an ais. You can use the -, -, and z-intercepts to help ou sketch a plane in space. equations of Lines and Planes in Space 191

20 Equation of a Plane in Space Given that A, B, C, and D are real numbers not all equal to zero and A is nonnegative, the equation of a plane in space is A + B + Cz = D Let us sketch the plane z = 1. z To find the -intercept, substitute 0 for and z. (0, 0, 3) O (0,, 0) + 6(0) + 4(0) = 1 = 1 = 6 The -intercept is the point (6, 0, 0). (6, 0, 0) To find the -intercept, substitute 0 for and z. (0) (0) = 1 6 = 1 = The -intercept is the point (0,, 0). To find the z-intercept, substitute 0 for and. (0) + 6(0) + 4z = 1 4z = 1 z = 3 The z-intercept is the point (0, 0, 3). (0, 0, ) z (0, 5, 0) To sketch the plane, plot the intercepts, draw the triangle connecting these points, and shade the plane. Remember a plane etends forever on all sides. Similarl, if ou have the graph of a plane, ou can work backward to find an equation from its intercepts. For eample, consider the plane with an -intercept of (10, 0, 0), a -intercept of (0, 5, 0), and a z-intercept of (0, 0, ). Each of these points gives values of,, and z that satisf the equation A + B + Cz = D. Substitute these values into the equation and solve for A, B, C, and D. (10, 0, 0) Substituting 10 for, 0 for, and 0 for z gives: A(10) + B(0) + C(0) = D 10A = D A = D 10 Substituting 0 for, 5 for, and 0 for z gives: A(0) + B(5) + C(0) = D 5B = D D B = 5 19 Unit 6 three-dimensional figures and graphs

21 Substituting 0 for, 0 for, and for z gives: A(0) + B(0) + C() = D C = D D C = When we put these values back into A + B + Cz = D we get: D 10 + D D 5 + z = D Because the plane does not pass through the origin, D is not zero, so we can divide both sides b D z + = 1. Once ou multipl the equation b the least common denominator of 10, ou have the equation in standard form, where A, B, C, and D are all integers z = 10 The plane graphed below has an -intercept at (4, 0, 0) and a z-intercept at (0, 0, 1), but it has no -intercept. This plane is parallel to the -ais. z (0, 0, 1) (4, 0, 0) You can still find an equation for this plane b eliminating the term from the equation A + B + Cz = D and solving as before. The equation becomes A + Cz = D. Substituting (4, 0, 0) and (0, 0, 1) gives: A(4) + C(0) = D 4A = D D A = 4 z A(0) + C(1) = D C = D So 4 + z = 1, or + 4z = 4, is an equation for this plane. Finall, consider the plane shown below. It is perpendicular to the z-ais, so it has neither an -intercept nor a -intercept, but it has a z-intercept at (0, 0, 5). (0, 0, 5) equations of Lines and Planes in Space 193

22 z z-trace In this case, eliminate both the - and -terms from A + B + Cz = D. This gives Cz = D. Substituting (0, 0, 5) gives 5C = D, which gives 5 z = 1. This is the same as z = 5. Equations of Traces You learned previousl that the intersection of the three aes creates three coordinate planes, the -, z-, and z-planes. A trace is the intersection of a plane with one of the coordinate planes. Remember that two planes intersect in a line. z-trace -trace An -trace is the line formed b the intersection of a plane with the -plane. The points on an -trace all have a z-coordinate of 0, so to find the equation of the trace, ou set z to 0 in the equation of the plane. A z-trace is the line formed b the intersection of a plane with the z-plane. The points on a z-trace all have an -coordinate of 0, so to find the equation of the trace, ou set to 0 in the equation of the plane. An z-trace is the line formed b the intersection of a plane with the z-plane. The points on an z-trace all have a -coordinate of 0, so to find the equation of the trace, ou set to 0 in the equation of the plane. Suppose ou want to find the equations of the traces for the plane given b z = 30. To find the -trace, substitute 0 for z (0) = = 30 The equation of the -trace is = 30. To find the z-trace, substitute 0 for. 5(0) + 6 z = 30 6 z = 30 3 z = 15 The equation of the z-trace is 3 z = 15. To find the z-trace, substitute 0 for (0) z = 30 5 z = 30 The equation of the z-trace is 5 z = Unit 6 three-dimensional figures and graphs

23 Parametric Equations and Equations of Lines Parametric equations epress variables in terms of another variable, called the parameter. The variable t is usuall the variable used to represent the parameter. In this book, parametric equations will be used to describe the equation of a line in three dimensions. Later in our math and science studies, ou ma use parametric equations to represent time and motion. Suppose we want to graph the line whose parametric equations are as follows: = 6t + 1 = t 1 z = 3t + 1 First, make a table and choose values for t. t z ordered triple 0 1 Second, substitute a value into each parametric equation to find the corresponding values for,, and z. When t = 0, = 6(0) + 1 = (0) 1 z = 3(0) + 1 = 1 = 1 = 1 When t = 1, = 6(1) + 1 = (1) 1 z = 3(1) + 1 = 7 = 0 = 4 When t =, = 6() + 1 = () 1 z = 3() + 1 = 13 = 1 = 7 t z ordered triple (1, 1, 1) (7, 0, 4) (13, 1, 7) Finall, plot the points (1, 1, 1), (7, 0, 4), and (13, 1, 7) and draw the line connecting them. equations of Lines and Planes in Space 195

24 z (13, 1, 7) (7, 0, 4) (1, 1, 1) Summar In space, an intercept is a point where a plane crosses an ais. The -, -, and z-intercepts can help ou sketch a plane. The equation of a plane is A + B + Cz = D, where A, B, C, and D are real numbers not all equal to zero and A is nonnegative. A trace is the intersection of a plane with one of the coordinate planes. Parametric equations epress variables in terms of another variable called the parameter. You can use parametric equations to graph a line in space. 196 Unit 6 three-dimensional figures and graphs