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Created for t⁴ by Thomas Sheppard and Steven Colgan Descriptive Geometry of Lines and Planes ©Galway Education Centre
Created for t⁴ by Thomas Sheppard and Steven Colgan
Descriptive Geometry of Lines and Planes
©Galway Education Centre
Table of Contents
Points
Planes
Inclined Planes
Oblique Planes
Quarter Sphere
Lamina
Rabatment 2
Cut Solids
Auxiliary Views
Skew Lines
Lines and Lamina
Rabatment 1
Inclination of a Plane
How to use this CD
First Angle Projection Intersecting Planes
Traces given a Lamina
Finding a Point
Finding a Line
Finding a Lamina
Lines
Contact Details
HOW TO USE THIS C.D.
To View a Slide Show Press the F5 key on the keyboard Or click the view “Slide Show” button at the bottom right of the screen Or select the “Slide Show” tab on the ribbon, then select “From Beginning” to view the presentation from the beginning
or “Current Slide” to view the show from the the current slide you are viewing
• Left clicking navigates you through the slide show
• All animations in the slides are activated using left clicks
• When a slide animation is completed, the next left click will take you to the next sequential slide
Left Clicking
• If you click on any of the orange text boxes (an example is shown below) in the table of contents slide, it will bring you to that particular section of the presentation
Action buttons
First Angle Projection
• There are three navigation icons in the bottom right of every slide
• The first of these is shown below
• If you click on this icon, it will bring you to the last slide viewed
• This may not necessarily be the preceding numbered slide
Slide Navigation
• There are three navigation icons in the bottom right of every slide
• The middle icon is shown below
• If you click on this icon it will take you to the table of contents slide
Slide Navigation
• There are three navigation icons in the bottom right of every slide
• The third icon is shown below
• If you click on this icon it will bring you to the next numbered slide
• This icon will skip all animations occurring on the current slide
Slide Navigation
• This is a button that appears in the top left corner of certain slides
• It is a hyperlink to an .avi file of the three dimensional animation on that particular slide or a recording of the animated line drawing on that particular slide
• This file is of higher quality than the animation in the powerpoint and can be paused, rewound and fast forwarded.
Animation button
• Some sheets contain embedded animations displaying key principles
• These animations will run automatically without any prompting
• To indicate an animation is present, these slides display a camera icon in the top left corner
• By clicking on this icon a media file of the animation opens in a seperate window
• This animation is of better quality than the embedded version and can be paused, fast-forwarded or rewound to specific moments in the animation for greater control
Animations
Sample of an embedded animation
Viewing AVI Animations
Play Fastforward/Rewind
Timeline Scroller
Set to Loop
Minimise, Maximise and Close screen
Fill screen
Viewing Flash Animations
Play
Fastforward Timeline Scroller Rewind
Navigating PowerPoint
To go to a specific slide in the presentation, right click your mouse
Navagating PowerPoint
To go to a specific slide in the presentation, right click your mouse A menu appears beside your pointer Move your pointer until the “Go to Slide” line is highlighted and allow the pointer hover in this position
Navigating PowerPoint
A new menu will appear Move your pointer onto this new menu and hover the pointer on the slide you wish to view The slide will highlight, and if this is the correct slide left click to confirm your selection
Navigating PowerPoint
You will then be taken directly to the selected slide.
• Often you will want to link one slide with another, such that if you click on a button during your presentation it will take you directly to a preset desired slide
• To do this a Hyperlink button can be created
• This is a button which when clicked during your presentation takes you directly to the preset slide
• Creation of a Hyperlink button is shown in the next slide
Creating a Hyperlink
Creating a Hyperlink
Firstly select the insert tab in the Powerpoint ribbon. This will open up the Insert options
Creating a Hyperlink
Firstly select the insert tab in the PowerPoint ribbon. This will open up the Insert options Now Click on the Shapes Icon
Creating a Hyperlink
Now select an Action Button to represent your Hyperlink
Firstly select the insert tab in the Powerpoint ribbon. This will open up the Insert options Now Click on the Shapes Icon
Creating a Hyperlink
Once you draw the Action Button the Action Settings box appears
Firstly select the insert tab in the Powerpoint ribbon. This will open up the Insert options Now Click on the Shapes Icon
Now select an Action Button to represent your Hyperlink
Creating a Hyperlink
Select Hyperlink to: Then select the drop down menu
Creating a Hyperlink
Select Hyperlink to: Then select the drop down menu
PowerPoint provides you with many items you can hyperlink to. But to hyperlink to a specific slide, select the “slide...” option.
Left clicking this opens up a menu with a list of slides
Creating a Hyperlink
Select Hyperlink to: Then select the drop down menu
PowerPoint provides you with many items you can hyperlink to. But to hyperlink to a specific slide, select the “slide...” option.
Select the required slide A Preview of the selected slide will appear
Confirm Selection by left clicking OK
Left clicking this opens up a menu with a list of slides
Creating a Hyperlink
This creates an active link within your presentation which can be accessed while delivering a presentation without having to search or escape from the presentation itself The link remains inactive during your presentation until selected by left clicking on the icon This links directly to your chosen slide
Creating a Hyperlink
To return to the slide you Hyperlinked from, left click on the “last viewed”slide icon
RECAP: FIRST ANGLE PROJECTION
First Angle Projection
POINTS
• A point has no width, height, or depth
• A points’ position can be defined using Cartesian coordinates (x, y, z)
• A point can represent:
– a specific position in space
– a point view of a line
– the intersection of two or more lines
What is a point?
LINES
• A line is of infinite length and has no thickness. It may be defined as the locus of a point as it travels along a straight path
• A line segment is a portion of a line – a line segment is commonly referred
to as a line
• A line represents the shortest distance between two points – i.e. to create a line two or more points
are needed
• A line can graphically represent: – the intersection of two plane surfaces – an edge view of a planar surface – a limiting element of a planar surface
• Lines are either vertical, horizontal, inclined or oblique. A vertical line is defined as a line that is perpendicular to the horizontal plane
What is a Line?
• A true length line is the actual straight-line distance between two points.
• In orthographic projection, a true-length line must be parallel to a projection plane
True Length of Lines
• A true length line is the actual straight-line distance between two points.
• In orthographic projection, a true-length line must be parallel to the projection plane it is projected on to.
• We see the true length of a line if we look perpendicular to this specific projection plane
True Length of Lines
True Length of Lines
• We see the true length of a line if we look perpendicular to the projection plane
• We see the line as a point if we look parallel to the projection plane (along the true length)
Viewing Lines
• We see the true length of a line if we look perpendicular to the projection plane
• We see the line as a point if we look parallel to the projection plane (along the true length)
Viewing Lines
Point View of Lines
• An inclined line is inclined at an angle to one of the projection planes and is parallel to another projection plane – These projection planes must
be at 90˚ to one another
• The true length of the inclined line is seen in one of the principal views – elevation, plan or end
elevation.
• The line is foreshortened in all other principal views
Viewing Inclined Lines
• An inclined line is inclined at an angle to one of the projection planes and is parallel to another projection plane – These projection planes must
be at 90˚ to one another
• The true length of the inclined line is seen in one of the principal views – elevation, plan or end
elevation.
• The line is foreshortened in all other principal views
Viewing Inclined Lines
Viewing Inclined Lines
• An oblique line is inclined at an angle other than 90˚ to all the planes of reference
• It is often called a skew line
• The line is foreshortened in all principal views
• The true length of the oblique line can be determined through the use of an auxiliary view or a rotation
Viewing Oblique Lines
• An oblique line is inclined at an angle other than 90˚ to all the planes of reference
• It is often called a skew line
• The line is foreshortened in all principal views
• The true length of the oblique line can be determined through the use of an auxiliary view or a rotation
Viewing Oblique Lines
Viewing Oblique Lines
PLANES
Line Generates a Plane
• A plane is an infinite flat surface
• A plane has no thickness
• A lamina is defined as a portion of a plane but is often referred to as a plane
What is a Plane?
Planes can be defined by:
1. Three non-linear points in space
2. Two intersecting lines
3. Two parallel lines
4. A line and point
• The point cannot be on the line
Planes
Planes are classified as: – Horizontal Planes – Vertical Planes – Profile Planes (End Vertical Planes)
• Perpendicular to the Vertical and Horizontal planes
– Planes of Reference • Horizontal, Vertical and Profile planes can be principal planes of
reference
– Simply Inclined Planes • Perpendicular to one principal plane and inclined to another principal
plane
– Obliquely Inclined Planes • Not perpendicular to a principal plane i.e. inclined to all principal
planes of reference
– Lamina • A lamina is defined as a portion of any plane mentioned above
Types of Planes
Planes of Reference
Vertical Plane
Horizontal Plane XY Line
Back to Basics!
• The Vertical Plane is where the Elevation is projected
• The Horizontal Plane is where the Plan is projected
• The XY line represents the Vertical Plane in Plan
• The XY line represents the Horizontal Plane in Elevation
The XY Line
First Angle Projection
Horizontal Planes
A plane inclined at 90° to the Vertical Plane
Vertical Planes
A plane inclined at 90° to the Horizontal Plane
End Vertical Planes
A plane inclined at 90° to the Horizontal Plane and 90° to the Vertical Plane
Inclined Planes
A plane inclined at 90° to one plane of reference and inclined at X° to the other (where X is not equal to
90°)
Oblique Plane
A plane inclined to both Planes of Reference
Note: the sum of the true angles of inclination of the plane to the Vertical and the Horizontal planes can
never be less than 90°
Sum of the inclinations cannot be less than 90°
Planes
Vertical Plane
Inclined Plane Oblique
Plane Horizontal Plane
End Vertical Plane
Traces
• Vertical and Horizontal Traces are lines of intersection between a plane and the Principal Planes of Reference
• The Vertical Trace of a plane is the line of intersection between the plane and the Vertical Plane – The Vertical Trace is seen in the Elevation
• The Horizontal Trace of a plane is the line of intersection between the plane and the Horizontal Plane – The Horizontal Trace is seen in the Plan
• The XY line represents the Horizontal Trace in Elevation
• The XY line represents the Vertical Trace in Plan
Traces of Inclined Planes
Vertical Trace
Horizontal Trace
A plane inclined to the Horizontal Plane
Traces of Oblique Planes
Vertical Trace
Horizontal Trace
INCLINED PLANES
• An Inclined Plane is inclined to one principal plane of reference and is perpendicular to another principal plane of reference
• The true inclination of an Inclined Plane can be seen in either the elevation, the plan, or the end elevation
• One trace is inclined at an angle to the XY line in either plan or elevation while the other trace is perpendicular to the XY line
Inclined Planes
Inclined planes
Vertical Trace
Horizontal Trace
A Plane inclined to the Horizontal Plane
Traces of an Inclined Plane
Vertical Trace
Horizontal Trace
A Plane Inclined at X° to the Horizontal Plane
X°
True Inclination to the Horizontal Plane
Vertical Trace
Horizontal Trace
Inclined Planes
A Plane inclined to the Vertical Plane
Vertical Trace
Horizontal Trace
Vertical Trace
Horizontal Trace
A Plane Inclined at X° to the Vertical Plane
X°
True Inclination to the Vertical Plane
Vertical Trace
Horizontal Trace
RABATMENT OF INCLINED PLANES
• To find the true shape of a surface cut by an inclined plane we can rabat the plane
– This means to knock down flat
• A rabatment shows us the true shape of the inclined plane and therefore the true shape of any surface contained within the inclined plane
• When rabating a plane we hinge the plane upon one of its Traces.
Inclined Planes
• Given the Traces of a simply inclined plane, rabat the plane about the Vertical trace
Question
• Given the Traces of a simply inclined plane, rabat the plane about the Horizontal trace
Question
OBLIQUE PLANES
• Oblique planes are inclined to both the Vertical and Horizontal planes
• The angles their Traces make with the XY line in plan and elevation are apparent angles and not the Oblique planes true inclination
• The sum of an Oblique planes true angles cannot be less than 90°
Oblique Planes
Oblique planes
Vertical Trace
Horizontal Trace
The Traces of an Oblique Plane
FINDING A POINT
Finding a Point
• Given the Traces of an Oblique plane and the plan of the point P, find the elevation
Question
P₁
P
P₁
P
• Given the Traces of an Oblique plane and the elevation of the point P, find the plan
Question
P₁
P
P₁
P
FINDING A LINE
• Given the Traces of an Oblique plane and the elevation of a line AB, find the plan
Question
B₁
B
A₁
A
B₁
B
A₁
A
• Given the Traces of an Oblique plane and the plan of the line AB, find the elevation
Question
B₁
B
A₁
A
B₁
B
A₁
A
FINDING A LAMINA
• Given the Traces of an Oblique plane and the elevation of a plane ABC, find the plan.
Question
B₁
B
A₁
A
C
C₁
B₁
B
A₁
A
C
C₁
• Given the Traces of an Oblique plane and the plan of a plane ABC, find the elevation
Question
B₁
B
A₁
A
C
C₁
A LAMINA AS A SECTION OF AN OBLIQUE PLANE
• As mentioned earlier a lamina is a section of a plane.
• This means that if we extend the lamina until it meets the Vertical plane and the Horizontal plane we shall find the traces of the plane that the lamina is contained on.
A lamina as a section of a plane
Given a Lamina find the Plane that contains it
• Given the plan and elevation a Lamina find the Traces of the plane that contains it.
Question
B₁
B
A₁
A
C
C₁
B₁
B
A₁
A
C
C₁
INTERSECTING OBLIQUE PLANES
• When two planes intersect they produce a line common to both planes
• This line is known as the Line of Intersection
Line of Intersection
Elevation of Line of Intersection
Plan of Line of Intersection
Line of Intersection
• The dihedral angle is defined as the true angle between two intersecting planes or lamina.
• It can be found by using either auxiliary projecton or rabatment method.
• In order to find the dihedral angle using rabatment it is first necessary to rebat a plane contining the true length of the Line of Intersection
Dihedral Angle
Dihedral Angle
Dihedral Angle
RABATMENT OF OBLIQUE PLANES
• We basically knock the Oblique/Inclined plane down flat, onto a principle plane
• When we rabat an Oblique Plane it the shows us the True Angle the Oblique plane makes with the Horizontal plane or Vertical Plane
• We rabat both Inclined and Oblique planes, to see the True Shape of the Cut Surface of an object.
Rabatment of planes
Rabatment of a Plane
Rabatment of an Oblique Plane
• Rabat the Oblique plane about the Horizontal trace
Question
P₁
P
P₂
P₁
P
P₂
AUXILIARY VIEWS
• When viewing an object in orthographic projection the projection plane is always constructed perpendicular to the viewing direction.
• If a view other than the standard plan, elevation and end view is required an auxiliary plane can be constructed to display the desired view.
• Auxiliary planes can be classified as either an auxiliary plan or an auxiliary elevation depending on the view it is projected from.
Auxiliary Views
• An auxiliary elevation is projected from an initial plan view.
• The height of the object relative to the horizontal plane remains constant with the previous elevation.
• The projection plane will always be perpendicular to the viewing direction
• A true view of a surface will always be seen when viewing perpendicular to the surface
Auxiliary Elevation
Auxiliary Elevation
True Surface
Auxiliary Plan
• An auxiliary plan is projected from an initial elevation view
• The distance of the object relative to the vertical plane remains constant with the previous plan
Auxiliary Plan
True Surface
Two Auxiliaries • When using auxiliary projection each auxiliary plane
will be perpendicular to the plane that went prior to it.
• In order to view an object from an oblique direction (that is a direction angled away from both the horizontal and vertical plane of reference) it is necessary to take two consecutive auxiliaries to achieve the desired view.
• To find the true shape of an oblique surface an initial auxilary must be taken parallel to the desired surface (this results in an edge view of the desired surface)
• Following this a second auxiliary is taken, viewing perpendicular to the found edge to give the true shape of the surface
Secondary Auxiliaries
True Surface
INCLINATION OF A PLANE
Background
• When dealing with true inclinations it is often useful to use the idea of a cone to determine a specific angle.
• A cone is a revolved solid, that is to say it can be generated by spinning an line or triangle of specific angle about an axis.
• The angle of the generator determines the base angle of the cone.
• Any plane laid against a cone will be inclined at the same angle as the cone
Line Generates a Cone
• The angles the Traces make with the XY line are the apparent angles
• To determine the true angles an Oblique plane makes with the Vertical or Horizontal plane, we slide/construct a cone underneath the plane.
• This is because any plane resting on a cone is inclined at the same base angle as the cone and vice versa.
• Therefore the base angle of the cone is the same as the inclination of the plane.
Inclination of an Oblique Plane
• If a plane is inclined at 60° to the Horizontal plane then we can slide/construct a cone of base angle 60° on the Horizontal plane and under the Inclined/Oblique plane to achieve the required plane
• If a to achieve the required plane
• plane is inclined at 30° to the Vertical plane then we can slide/construct a cone of base angle 30° on the Vertical plane and under the Inclined/Oblique plane
Oblique Planes
Inclination of a Plane
Cone with base angle the same inclination as the Plane
X˚
Inclination of a Plane
True Inclination to the Horizontal Plane
X°
P₁
P
Inclined to the Vertical Plane
X˚
True Inclination to the Vertical Plane
X°
P₁
P
Inclination of a Plane
• If you are given a trace and told the inclination to a plane of reference then you can construct a cone to determine the unknown trace
• Given the Vertical trace, and a true inclination of 70˚ to the Horizontal plane. Draw the Horizontal trace
Question
True Inclination to the Horizontal Plane
True Inclination to the Horizontal Plane
X° X°
P₁
P
• Given the Horizontal trace, and a true inclination of 55˚ to the Vertical plane. Draw the Vertical trace
Question
True Inclination to the Vertical Plane
X°
P
P₁
• To determine the true angles an Oblique plane makes with the Vertical or Horizontal plane auxiliary views can be used
• This results in an edge view of the plane and an edge view of the cone.
• Typically you are asked to locate a view which shows the true inclination of the Oblique Plane to a specified plane of reference
Oblique Planes
• Find a view which shows the true inclination of the Oblique Plane to the Horizontal Plane
Question
P₁
P True Inclination to the Horizontal Trace
X°
• Find a view which shows the true inclination of the Oblique Plane to the Vertical Plane
Question
Distance
X°
P₁
P
Auxiliary views
• Note that the entire cone which the plane rests upon need not be drawn.
P₁
P
Auxiliary elevations
• When we get an Auxiliary elevation of an Oblique plane, we see the Horizontal trace as a Point, and the Vertical trace as an Edge/Line
• This means we are seeing and an Edge view of the Oblique plane.
– We use Auxiliary Elevations to see where an object is cut by an Oblique plane.
– We use Auxiliary Elevations to rest an object on an Oblique plane
Auxiliary Plans
• When we get an Auxiliary Plan of an Oblique plane, we see the Horizontal trace as a Edge/Line, and the Vertical trace as an Point.
• This means we are seeing and an Edge view of the Oblique plane.
– We use Auxiliary Plans to see where an object is cut by an Oblique plane.
– We use Auxiliary Plans to rest an object on an Oblique plane
To determine the true inclination of an Oblique plane to the Vertical plane
• Look along the Horizontal trace (parallel to the trace)
• Draw an XY line perpendicular to this line • Pick a point on the Vertical trace • Find this point in plan (it is on the XY line) • Bring this point into the auxiliary elevation an
mark off its height (the distance from the elevation)
• Joint this point to the intersection of the X1 Y1 and the extended Horizontal trace
Oblique planes
To determine the true inclination of an Oblique plane to the Horizontal plane
• Look along the Vertical trace (parallel to the trace)
• Draw an XY line perpendicular to this line • Pick a point on the Horizontal trace • Find this point in elevation (it is on the XY line) • Bring this point into the auxiliary plan an mark off
its height (the perpendicular distance from the plan to the XY line)
• Joint this point to the intersection of the X1 Y1 and the extended Vertical trace
Oblique Planes
QUARTER SPHERE
Oblique Planes
• As mentioned earlier the angle a trace makes with the XY line is the apparent angle
• This does not give the true inclination of the Oblique plane
– This does not apply to an inclined plane
• If we are given the true inclination of Oblique plane to the Vertical plane and the Horizontal plane then we need to use the quarter sphere method
• We need to create a cone whose base angle and inclination is equal to the Oblique planes inclination to the Vertical plane
• And we need to create a cone whose base angle is and inclination is equal to the Oblique planes inclination to the Horizontal plane
• However these two cones are not related to one another • To do this we create a sphere of any size and place the
cones on top of the sphere • This creates a proportional ratio between the cones
– i.e. as the sphere gets smaller then the cones get smaller and as the sphere gets bigger then the cones get bigger
• This is known as the quarter sphere method – This method is used when true inclinations to the Vertical and
Horizontal plane are given not apparent ones
Oblique Planes
Quarter Sphere
Quarter Sphere Method
• Draw the Traces of an Oblique Plane which has a true inclination of 60° to the Horizontal plane and 45° to the Vertical Plane
Question
CUT SOLIDS
Cutsolids
• Planes are often used as constuction entities for a drawing, however it is often useful to imagine a plane cutting through the body of an object removing a section of the solid.
• Such a process results in part of the solid being cut away, often to illustrate internal detail not readily seen or to create a new solid object
Object
• Planes are often used as constuction entities for a drawing, however it is often useful to imagine a plane cutting through the body of an object removing a section of the solid.
• Such a process results in part of the solid being cut away, often to illustrate internal detail not readily seen or to create a new solid object
Cutsolids
Cutting Plane
Cutsolids
• Planes are often used as constuction entities for a drawing, however it is often useful to imagine a plane cutting through the body of an object removing a section of the solid.
• Such a process results in part of the solid being cut away, often to illustrate internal detail not readily seen or to create a new solid object
• The resulting surface created by the plane is known as the cut surface
• There are various methods to achieve the true shape of this surface. The most commonly used of these methods are the Auxiliary view method and the rabatment method
Cutsolids
Cut surface
Cut surfaces
• In order to view the actual intersection of the plane and the object we need to view the cutting plane as an edge
• For an inclined plane, we simply take the points (Where the trace cuts the object) up from our plan or down from our elevation.
• For an oblique plane we must first find an auxiliary view in which the cutting plane is viewed as an edge
True shapes of cut surfaces
• Having found the intersection of the cutting plane and the solid it is often useful to determine the true shape of the cut surface.
• This can done by either rebating the cut surface such that it is parallel with a primary reference plane
• Or by creating an auxiliary plane parallel with the cut surface
Inclined Plane
True Surface
True Surface of a Cut Solid
True Shape of Cut Surface
Auxiliary Method
True Shape Of Cut Surface
Alternative Cut Surface
SKEW LINES
• Because a skew line is not orientated parallel to either of the primary projection planes both the plan and elevation of the the line will be forshortened.
• In order to view the true lenght of a line the line must be orientated parallel to a given plane. There are two methods to do this
1) The creation of an auxiliary plane set parallel to the skew line
2) Rebating the plane such that it is now orientated parallel to one of the principal projection planes
True Length of a Skew Line
True Length using Auxiliary Method
A1
B
A
B1
A2
B2
Rabatment of Skew Lines
A1
B1
A
B
B
B1
True Length
Point view of a skew line
• To view a line as a point view we must look along the lenght of the line.
• To do this using orthographic projection we must first see the true length of the line
• A point view can then be achieved by looking parallel to the true length line and projecting the line on to a perpendicular plane
Point View
A1
B
A
B1
A2
B2
A3,B3
Point view of line
Intersecting Lines For two lines that intersect the point of contact will allign in both plan and elevation
Skew Lines If the two lines are skew lines and don’t intersect the the apparent point of contact will not allign in plan and elevation
C1
D
B
A C
A1
D1 B1
C1
A
C
D
B
A1
D1 B1
Lines as part of a plane
• All lines can be contained on an infinite number of panes.
• These planes can all be rotated about the line
• All lines can be contained on a vertical plane
• Only horizontal lines can be contained by a horizontal plane
Planes Containing Lines
Common planes for two skew lines
• As just seen there are an infinite number of planes with infinite number of inclinations that can contain a line.
• When dealing with two skew lines however there is only one plane with an inclination that can contain both lines using two identical planes
• To find this plane three points must be found on the plane that relate to both lines.
• One line contains two points so to relate the second line to the first a parallel line can be drawn on one end of the first line with inclination of the second. This will give three points on this common plane.
Finding the edge view of the parallel planes
• Having three point on the plane will create the desired plane but to view the panes as an edge view an auxiliary must be taken
• To find the correct direction to view the planes construct a horizontal plane in Elevation
• Where this intersects the first skew line and the constructed parallel line are two points on the line of intersection between the containing plane and the horizontal plane.
• Project these points to the plan view to get a true length of this line of intersection.
• Looking along the true length of a line on a plane will result in viewing the plane as an edge view thus the direction for the required auxiliary is found.
Skew Lines Parallel Planes
C
B A
D
B D
C A
C
D
B
A
Shortest Perpendicular Distance
B2
C
B A
D
B1 D1
C1 A1
C2
D2
D3 A3
C3 B3
A2
Alternative Perpendicular Distance
C
D
D1 A1
C1 B1
B
A
C2
D2
B2
A2
C3,D3
A3
B3
Shortest Horizontal Distance
A3
C
B A
D
B1 D1
C1 A1
C2
D2
B2
A2
Direction of Sight is parallel to Auxiliary X1,Y1 line
A
B3
D3 C3
Shortest Distance at an Angle
B2
A1
C
B A
D
B1 D1
C1
C2
D2 A2
Direction of Sight is set to the given angle for shortest distance
C3
D3
A3
B3
LAMINA
Lamina
• A lamina can be defined as a section of a plane
• It can be any shape or size but must have at least three points.
• The true inclination of the lamina will be the same as the plane it is a section of.
• A lamina has no thickness
Finding an Edge view of a lamina
• To find the edge view of a lamina an auxiliary view must be taken.
• The two most common ways to do this are by:
1. Finding a point view of an edge on the lamina
2. Creating a horizontal cut and looking along the intersection of the Horizontal plane and the lamina
Edge View of a Lamina Auxiliary Method
B1
C1
A1
B
A
C
B2
A2
B3,C3
A3
C2
True length of Edge
Point view of Edge Edge view
of surface
Finding the Edge view of a lamina
A1
C2
A2
B
C
A
C1
B1
B2
True Length of Horizontal Line
Horizontal Line
Edge View of the Plane
Point view of True length line
Line of Intersection
• When one plane pierces another a line common to both planes will be created where they both pierce.
• This is known as the Line of Intersection
• To find this line either of two methods are commonly used.
1. Viewing one plane as an edge view
2. Using horizontal cutting planes to view common points on both planes
Intersecting Planes
Line of Intersection Edge View Method
A1
F
C
D
A
F
D1
E1
C1
B1
A2
B
B2
C2
E2
D2
F2
E
Line of Intersection Horizontal Cut
A1
B1
C1
E1
F1 D1
D
E B
F
C
A
A1
B1
C1
E1
F1 D1
D
E B
F
A
C
A1
B1
C1
E1
F1 D1
D
E B
F
A
C
A1
B1
C1
E1
F1 D1
D
E B
F
A
C
Dihedral angle of two lamina
• As seen previously The dihedral angle is the true angle between two planes
• In order to view the dihedral angle the Line of Intersection must be viewed as a point view.
• This can be done using two auxiliaries views.
• The first auxiliary is viewed looking perpendicular to the L.O.I giving a true length.
• The second auxiliary is setup looking parallel to this true length line giving a point view of the line and consequently the dihedral angle of the two lamina
Dihedral Angle
A1 D1
A
B1
C1
B
C
D
A2
C2
D2
B2 Dihedral Angle C3B3
D3 A3
LAMINA AND LINES
To Determine the Shortest Perpendicular Distance from a Point to a Lamina
• Find the Shortest Perpendicular Distance from the Point P to the Lamina ABC
Question
A₁
C₁
B₁
P₁
A
C
B
P A₂
C ₂
B ₂
P ₂
To Determine the Shortest Horizontal Distance from a Point to a Lamina
• Find the Shortest Horizontal Distance From the Point P to the Lamina ABC
Question
A₁
C₁
B₁
P₁
A
C
B
P
A₂ C ₂
B ₂
P ₂
To find a Line from a given point to a lamina that is Xmm long and is Y° to the Horizontal Plane
• Draw a line from point P which will though the lamina ABC at a distance of 90mm from P and is inclined at 45° to the Horizontal Plane
Question
A₁
C₁
B₁
P₁
A
C B
P
A₂
C ₂
P ₂
To Determine the true inclination from a Line to a given Lamina
• Find the true inclination of the line PC to the Lamina ABC
Question
A₁
C₁
B₁
P₁
A
C
B
P
A₂
C ₂
B ₂
P ₂
A₃
C ₃
B ₃
P ₃
How to draw a line from a given point that is Xmm long and parallel to the vertical plane an a given lamina
• Draw a line from the point P which is 50mm long, and parallel to the lamina ABC ad Parallel to the Vertical Plane
Question
A₁
C₁
B₁
P₁
A
C
B P
A₂ C ₂
B ₂
P ₂
How to draw a line from a given point on a lamina that is inclined to the edge of the lamina at a specified angle
• Draw a line from the point A which will make and angle of 50° to the edge BC on the lamina ABC
Question
X˚
A₁
C₁
B₁
A ₂
C ₂
B ₂
A
C
B
C₃
A₃
B₃
How to draw a line from a given point on a lamina that is inclined to the edge of a lamina at a specified length
• Draw the projections of a line on the lamina ABC which is 30mm long, starts at B and ends on the edge AC
Question
A₁
C₁
B₁
A ₂ C ₂
B ₂ A
C B
C₃
A₃
B₃
CONTACT DETAILS
If you have any queries, suggestions or comments regarding this teaching resource
please contact Tom Sheppard or Steven Colgan at:
Contact Details
