Transcript for:
Lecture 1: 3D Computer Vision

hello everyone welcome to lecture 1 of 3d computer vision and today i'm going to talk about 2d and 1d projective geometry hopefully by the end of today's lecture you'll be able to explain the difference between euclidean and projective geometry in particular we'll look at the familiar cartesian 2d coordinates to describe the euclidean geometry and we'll look at how to make use of what we call the homogeneous coordinates to describe the projective geometry in particular we'll use homogeneous coordinates to represent points lines and conics in the 2d projective space in today's lecture we'll also look at the duality relationship between lines and points and conics and dual conics on a 2d plane finally we'll apply the 2d hierarchy of transformation on points lines and conics in today's lecture of course i didn't invent any of today's material i took most of the content of today's lecture from the textbook written by richard hartley and andrew zizermann multi-view geometry in computer vision in particular chapter 2 for today's lecture i also took some of the materials from the textbook written by mahi an invitation to 3d vision chapter 2. i strongly encourage all of you to take a look at these two chapters in the two textbooks after today's lecture to reinforce your idea on today's concept projective transformation is actually a part of our life is actually a very strong part of our life and this is because we live in a 3d world where everything follows the euclidean geometry that can be described as a cartesian coordinate but the moment that we open our eye and see the world we are actually seeing the world through projective transformations this is because light that is cast on any objects in the 3d world it's reflected into our eye and it converges into our retina via the concepts of projective transformation and what's interesting here is that the projective transformation causes geometrical changes to the 3d world that we are living in and the images that is formed in our eye that is sent into our brain in particular we can see that from this for example in this picture here we can see that parallel lines in the 3d world this pair of lines is in fact a paler line which should not meet in the 3d world that we are living in in particular this particular building over here so what's interesting here is that when it's projected onto an image it undergone some form of projective transformation and we can see that the pair of line they actually meet at a certain point here at a finite point in the image this is the same thing when we are looking at scene via our eye because our eyes are actually the most powerful form of cameras that is uh available and uh it functions the same way in this particular case here where i have a digital camera where the lights are got projected into the photo sensors in my camera over here and we can further see that rectangles that are in the real world scene for example the window over here because of the projective transformation when this the light ray is being projected from the scene into the camera we can see that the windows that is being formed on the image no longer there are no longer rectangles over here in fact it becomes some form of parallelogram over here and another example here would be the circle this clock over here this clock phase over here in the 3d world in the euclidean world that we're living in it's actually a perfect circle but when it is projected onto the image after projective transformation we can see that the geometry circle is not preserved it becomes an ellipse in this case over here and we saw from the previous slide that certain geometrical properties they are not preserved by projective transformation in particular we saw three examples a circle may not appear as a circle anymore it becomes an ellipse and a parallel line they will not uh they will not stay parallel they actually meet at a certain finite point when it is being projected into the image and as well as a rectangle may appear as a parallelogram when it's undergone some form of projective transformation and in fact if a closer look at the images or a closer look at the world through our eyes it's not difficult to realize that the in fact the angles the distance and the ratio of distances are all not preserved for example in this case here i have two parallel lines in the scene uh these are supposed to not form any anger at all but when it's projected onto the image you can see that this is what is going to happen it's going to meet at a certain point and it's going to form some anger even lines that are not parallel for example they form some already form some angle of data over here when it's being projected onto an image we can see that this angle of theta it can actually change to data prime which is not the same as what is what you see uh what or what it really exists in the 3d world and the distance here could refer to the distance between two points two real points in the 3d world so suppose that this distance here is denoted by d over here when it's projected onto an image we can see that this d over here between these two points it becomes d prime they will not never be the same or it's unlikely that they are going to be the same as well as the distance between the two end points of a line or the perpendicular distance of two lines suppose that i'm looking at this distance denoted by d over here that's formed by two lines and the projection to an image would cause this d over here to change into d prime for example as well as the ratio of the distances suppose that i have two distances that is formed by this uh four points over here uh this let's call this d1 and this d2 over here and the ratio between these two distances is denoted by d2 over d1 in the 3d world but when it is projected onto an image we can see that this d1 becomes d1 prime and d2 here becomes d2 prime that means that d2 prime and d1 prime will not be equals to the ratio of d2 and d1 due to the effect of projective transformation we shall see that all these analysis all these mathematical or geometrical analysis can be formalized using the homogeneous coordinates that we are going to learn in a in a while so we have seen earlier that most of the geometrical property changes uh when it undergoes some form of projective transformation so the question here would be to ask ourselves on what's the a good way to quantify or what's a good way to study this projective transformation unfortunately a property that is preserved in projective transformation is what we call the straightness property and this becomes the most general requirement on the projective transformation is in other words we say that the straight line or straightness property stays invariant after certain form of after the projective transformation we can see from the image earlier on that in that case any straight line in the scene is actually still projected into a straight line after it has undergone some form of protective transformation despite that the anger the ratio of the distance and the distance may change but the straight line will still remain as a straight line after some form of projective transformation and the thought here is that we may actually make use of this invariant property of the straighteners uh to define a projective transformation or to describe the whole set of projective geometry that preserves the straight line this means that mathematically we can formulate some form of projective mapping or some form of mapping that maps from one projective space so this is a projective space that maps from one projective space to another projective space over here for example in the 2d projective space which i'm going to denote as p2 over here so uh but in this case over here since we observe that straightness is being preserved what it means is that we want to define a projective transformation which i now call temporally as h over here that maps any straight lines in in one of the scene into another scene such that the straightness property here will also be preserved and more generally we will study the geometric properties that are invariant with respect to projective transformations in projective geometry so straightness or straight lines preservation of the straightness in the projective transformation is one of the property will throughout these lectures we'll look at some other form of invariance geometrical invariance they are invariant to a certain form of projective transformation in projective geometry so so far what we are familiar with in geometry is what we call the euclidean uh geometry and this is an example of a synthetic geometry that we all have learned since our primary school days when we are taught geometry and mathematics this is also making use of what we call the exomatic methods and it's related tools to describe geometry or to solve any problems in geometry for example if i want to solve the a triangle a right angle triangle and i solve the angle over here we can make use of compass and rulers to actually solve this particular problem we can measure the angles between any two lines in this triangle using a compass or we can just simply measure the distance of this triangle the edges of this triangle using a ruler and computes the angle using the cosine rule or the sine rule so uh this is the most standard form of euclidean geometry that every one of us learned in school and in fact euclidean geometry has also a very long history it was actually invented by the greeks in very early days as depicted in this particular painting over here by rafael in the 1500s we can see that in this painting over here there's a group of philosophers or there's a group of mathematicians trying to solve some form of euclidean geometry problem by making use of the compass and the straight edge to solve this particular problem and what we're going to look at in this module over here is what we call the projective geometry and instead of making use of exomatic methods and the related tools such as compass and straight edges to solve the problem we are going to make use of mathematics we are going to make use of coordinates and in in particular algebra or linear algebra were to solve problems in projective geometry and we can see that we will learn in the later part of the lectures how we can convert projective geometry representation in particular homogeneous coordinates into the euclidean geometry or the euclidean space so euclidean space are usually represented by the cartesian coordinates and the what we call projective geometry since it's making use of math and coordinates and linear algebra to describe projective geometry and to solve problems in projective geometry we'll call this also the analytical geometry we shall see that one of the most important results from projective geometry is that the geometry at infinity can be nicely represented so in euclidean space there's no way to represent any points that that occurs at infinity or any lines that occur at infinity or any planes that occurs at infinity and in fact the seemingly only way to represent this line's point or planes at infinity in euclidean geometry is to make use of the infinity representation itself which it ends up to be an exceptional case in mathematics when we do any mathematics operations infinity it's often unrepresented or it ends up to be a exceptional case in euclidean geometry but we'll see that by making use of homogeneous coordinates in projective geometry the infinite points lines and planes can be represented elegantly using just a set of numbers which where we do not need to treat them well as a separate case so we'll make use of homogeneous coordinates to describe the projective space as we have learned in our high school mathematics or geometry that in the euclidean space which we denote as e over here can be easily represented by a set of cartesian coordinates which are living in the real number space so for example in the 2d space 2d euclidean space this is usually represented using 2d real numbers in the cartesian coordinates as compared to in the projective space in this case over here if we are talking about the 2d projective space then this would be represented by a set of homogeneous coordinates which are a set of three numbers they are also taking a form of real numbers but in this case this will be in the r3 space where we are talking about three numbers over here we'll see that a point in the homogeneous coordinates can be represented with using these three numbers k x k y and k respectively so this form these three numbers over here it forms a point in the homogeneous coordinate or it represents a point in a homogeneous coordinate that corresponds to kx divided by k and ky divided by k where k over here can be cancelled off to get x and y in the cartesian coordinate so here in this particular statement over here what we have seen is that given a homogeneous coordinate we can easily convert it into back into the euclidean space or into the cartesian coordinates over here by simply dividing the first two coordinates of the homogeneous coordinates with the last entry in the homogeneous coordinates over here and it's also interesting to note that uh k x k y and k is equivalent for all uh forms of k so we can think of it this way that the k here is just a scalar value that is multiplied by the homogeneous coordinates of x y and one over here so for all case over here this particular representation of homogeneous coordinates represents the same point of x and y in the cartesian coordinate space now we can also use the homogeneous coordinates representation these three numbers to represent a point at infinity a 2d point and infinity by simply setting the third value here k to be equals to 0 and we can see that this is an elegant form of representing points 2 or 2d points at infinity because it doesn't require the exceptional representation of infinity or the concept of infinity over here all we need to do is just to set the last value over here k to be equals to zero and we can see this clearly when we convert this set of points over here this homogeneous coordinate x y and zero over here into the cartesian coordinates by simply dividing x over zero and y over zero following the rules that we have seen earlier in the first statement over here and this will end up to be infinity infinity where uh in the cartesian space which actually represents a point at infinity but it's actually very inconvenient to represent infinity in the cartesian coordinates directly so here homogeneous coordinates offers us a more elegant way of representing points at infinity and generally the rn euclidean space can be extended to a pm projective space as homogeneous vectors as we'll see in this particular lecture we can see that this example over here for a point we can see that for a euclidean space of r2 represented by r2 space it can be actually extended to a projective space of p too easily by just dividing the values of k the third coordinates here and we can easily convert euclidean space to projective series and vice versa we will see in the next lecture that this can also be done in the r3 space with some difference from exceptional properties that we will see that arise from a higher dimensional space over here compared to the 2d space in particular the hierarchy of transformation and pictorially we can see that the homogeneous coordinates a point in the homogeneous coordinate can actually be represented by array as shown in this diagram over here so as what we have mentioned earlier on that any points on the homogeneous coordinates can be represented by these three numbers over here k x k y as well as k over here and this k is a scalar value that is multiplied by x y and one over here where for any k or for all the case uh this over here these three numbers over here would be the same uh 2d point in the cartesian space and what we can see here is that in this case here since the k over here is a scale what this means is that x y and 1 and 2x 2 y and 2 as well as all the way until k x k y and k they all represent the same point and if we were to join all these points together we can see that it actually forms a projection it actually forms a ray that joins out all these points over here that represents the same 2d point in the cartesian space and hence we can see that we can conclude that the homogeneous coordinates of a point or representation of a point is actually equivalent to array over here if we were to plot it up in the cartesian space of x y z coordinates over here so we can also look at another example over here suppose that there's another point over here which is actually given by k x prime k y prime as well as k over here we can see that this particular point is equivalent to on this particular plane over here is equivalent to 2x prime 2y prime as well as 2 over here and there is also a corresponding point of x prime y prime as well as one over here so if we were to join all these three points up then this will form a light array that passes through the all the sets of points and it's also interesting to note that it's not necessary that k takes an integer value so k need not be just an integer value of 1 2 3 4 and so on so forth it can actually be any real number so k over here can actually be any real number where it can actually be equals to 1.1 or 1.23 and so on so forth and in this diagram over here we can also see that based on our definition of a point at infinity it was given by x y and 0 as defined in the previous slide over here where x and y can take any value so what this means is that the all the points that are or any vectors on this that is lying on the x y plane on the x y plane over here they all correspond to points at infinity and uh we will also look at the uh what we call the line at infinity which we will represent it as l infinity uh later on that this is actually equals to 0 0 1 we'll see how to derive this particular representation in a few slides time and what's interesting here is that this l infinity is actually corresponding to the any point on the z axis over here since it's 0 0 1 and it's interesting to note that the origin of the cartesian coordinate or of 0 0 0 over here is undefined so this point is actually undefined in homogeneous coordinates we do not use this representation at all in hormone genius coordinate so when i look at the homogeneous notation for lines on a plane which means that we are looking at the 2d line and the incidence relations between the lines and the points uh which we will also from here we will also look at how the the to derive the duality relation between lines and point we will realize that the lines and the point representation on homogeneous coordinates are actually interchangeable and in the pictorial form that we have seen earlier we saw that a point in the homogeneous coordinate is actually represented by a ray represented as three numbers kx ky and k well it's going to be the same for all case that's why the point is actually represented as a ray in the projective space in the homogeneous coordinates and we will see that a line in the on the plane a 2d line on a plane over here which we denote as l over here it actually is represented by a plane because it projects to become a plane in the to uh in the 2d projective space and in this case here we can see that a plane can be represented by its normal vector so what happens here is that line the representation of line geometrically it's actually the represented by the normal vector of the plane that is uh projected from this particular line over here now we all know that a line in a plane which is a 2d line can be represented as this particular polynomial equation over here ax plus b y plus c equals to zero this simply represents a line in the x y plane in the uh cartesian coordinate and this is the equation of a x plus b y plus c or if we were to manipulate this particular equation a little bit we can see that this simply becomes y equals 2 minus a divided by b x minus c or simply y equals to m x plus c which is the familiar equation of lines that we all have learned throughout our high school mathematics or even junior high school mathematics and we know that the different choices of the parameters of x y or the coefficients of x y which is a b and c will give rise to different lines because a and b here controls the gradient of the of this line on how steep this line will be on the x y plane over here and c here would be the intersection of the line with our y axis over here thus a line can be represented naturally by the vector these three numbers a b and c without having the need to the concern about x and y over here and hence in homogeneous coordinate representation of lines we'll just simply make use of the coefficients to the line equation over here in 2d abs and c to represent the lines in homogeneous coordinates so we can see that there is a similarity in the representation with the homogeneous coordinates of points which is also represented by three numbers over here and but physically or geometrically it actually means different things so in the case of the uh lines or in the case of the points is kx ky and uh k over here and this is actually representing the all the points that are in the array that is formed by the projection onto the origin and the intersection with all these planes over here which determines the cave value would determine the point in the cartesian coordinate and in this case here the line is also represented by a three number but in this case a b and c over here simply represents the coefficient of the lines and it's important to note that the correspondence between the homogeneous representation of a line as a b c a 3 vector of a v c here is not uh oh there's no one to one correspondence between this homogeneous representation and the usual vectors that we known of in the cartesian space of a b and c transpose over here this is because the lines are represented by this polynomial equation of a x plus b y plus c equals to zero and we can see that in in the homogeneous representation or in the line representation this polynomial uh equation over here when multiplied by a scalar value of k over here it's going to be the same uh line uh it's going to represent the same line because this k here can be simply factorized out into k multiplied by ax multiplied plus b y plus c equals to 0 and this k can be cancelled off so regardless of the value of k here it's always going to represent the same line but in the case of a a vector if we were to write the vector as k a k b and kc here these are going to represent the different lines for or different vectors different three vectors in the cartesian space for different values of k and but in this case here in the homogeneous representation here we say that a b c the three vector of a b c and the three vector of k a b and c transpose represents the same line for any non-zero k and h therefore what this means is that they are equivalent class and note that the vector of 0 0 0 as i mentioned earlier when we look at the diagram that represents the projective transformation the projective space it does not correspond to any line neither does it correspond to any point at all so this is what we call the singular point where it doesn't represent anything at all in the homogeneous coordinate now we'll see the incidence relation so incidence relation means that how a point and a line coincide or intersects with each other so in this case here it would be a line and we will look at the relation or the mathematical formulation in homogeneous coordinates form on how to represent the incidence relation where a point sits on a line the point let's denote it as x and the line let's denote it as l which is given by these two values over here and we say that a point lies on a line if and only if the it fulfills this particular equation in the cartesian space this is the equation the line equation that we all are familiar with in cartesian space and in fact a b and c here represents the parameters or the coefficients of the line and x and y here simply represent all the points that are in this line when we plot it out in the cartesian space of x and y over here so we can rewrite this form of the equation the polynomial equation into a matrix multiplication or vector dot product over here we can see that it becomes x y one uh multiplied by a b and c transpose and interestingly this x and y and 1 over here it becomes the homogeneous representation the homogeneous coordinates of a point of a point x y in the cartesian space and c transpose over here simply cause uh coincide with the what we have defined earlier to be the homogeneous coordinates of a line represented by l hence we can simply rewrite this into x y one multiplied by l which is the vector the three vector that represents the the the line and according to this particular polynomial equation that represents the line we can equate this product over here in to be equals to zero so similarly for any constant non-zero k this relation the incidence relation will always hold true and in this case here we can say that for any point represented as the homogeneous coordinate k x k y and k multiplied by a b and c it will always gives us 0 over here so it's going to be the same if we even if we were to scale l as what we have seen earlier that this is going to be scaled by another scale where we can factorize out say for example k prime so we can see that k and k prime are going to cancel out when we equate it to zero and at the end of the day is still this particular equation that represents the incidence relation between a point and a line hence the representation that we have described earlier of kx ky and k in 2d in the projective 2d space for varying k value is a valid representation of a point which is in the cartesian coordinate space and this actually also proves that or it actually shows the validity of our definition of the point in the projective space using homogeneous coordinates and we can see formally this would be the representation of any point in the homogeneous 2d space x1 x2 and x3 a 3 vector over here and we say that it's in a projective it's in a 2d projective space which is equivalent to the cartesian coordinates of x1 divided by 3 and x2 divided by 3 transpose in the cartesian coordinate space so in this case here this is the our projective space and this is our cartesian coordinate space more formally the point x lies on the line if and only if the dot product of the point and the line is equals to zero as what we have seen earlier so this dot product over here simply expressed out to be ax plus b y plus c equals to zero and we simply factorize the this into a dot product as written in this formal form over here so this is the first example that we see that how linear algebra because this is the dot product is from linear algebra how linear algebra can be used to describe the incidence relationship between a point and a line in the projective 2d space and note that the expression of x transpose l is just the inner or scalar product or simply call the dot product or as i've mentioned several times and the dot product is equivalent regardless of how we sort the order of this x and l over here because we can simply see that you will still evaluate into this form over here and we shall see later that because of this swapping over here we can end the same representation with the free vector for a point and a line in the 2d projective space hence we can see that the row of the point and the line can be interchanged which we'll look at in more detail when we talk about the duality relationship and now the degrees of freedoms of a point is 2 degrees simply because we saw that there's x and y coordinates where it can be changed x and x and y respectively has one degree of freedom and the line also has uh 2 degree of freedom in the 2d projective space because there are two independent ratios of a this to b is to c and this can actually be made more clear when we by the example that i've shown earlier that we can actually rewrite a x plus b y plus c equals to 0 into y equals to minus b or a divided by b x minus c which is simply equals to m x plus c so in this case here we can see that there are only two parameters that matters which is m and c which is the gradient and the y-intersection of the line so m over here is actually formed by the ratio so uh it should be b divided by c over here so because b is a coefficient of y and it should be also divided c should also be divided by b so we can see that in this case here is the ratio that matters a divided by b and c divided by b that matters in the line hence a line also has two degrees of freedom in the projective 2d space now the intersection of two lines in the 2d space will always give rise to a point because any two lines are always going to meet at a certain point and we will see that include this includes parallel lines which we will assume that they or mathematically define them to intersect at infinity at a point at infinity and the formerly using the homogeneous coordinates representation two points always or two lines always intersect at a point would be given by the cross product of two lines suppose that we have two lines l and l prime the intersection which we denote as x over here would be given by the cross product of l cross by l prime over here and geometrically we can see that uh since i have mentioned earlier by showing the diagram that a line l in projective space is actually represented by a plane it because it projects to a plane uh and uh all the lines on this plane actually represent the all the parallel lines in this plane that lies on this plane it actually represents the same line because of the regular scaling that can be cancelled out as what we have as i have mentioned earlier that k a x plus b y plus c must be equals to 0 and k here doesn't matter and we can see that if we were to have two lines which we denote as l and l prime over here they both back project to a plane and what's interesting here is that on the 2d line on the two these two are 2d lines and on the 2d plane the intersection of these two lines will always be at a point over here which actually uh in the homogeneous representation homogeneous coordinate representation is denoted by this ray over here so this particular what this means is that the cross product when we take the cross product of these two lines over here we will get this point and in the homogeneous coordinate this is equivalent to the this particular ray or vector over here the three vector that we use to represent the homogeneous coordinate and uh a further description of this would be that the line here the geometric intuition of the line since it back projects to a plane we can also see it as the the vector over here the normal vector of these two planes respectively that represents the line of l and l prime over here and the cross product using the right hand rule over here the cross product of this two vector actually gives a octagonal vector that is uh in this direction over here which is directly corresponds to the point of x hence a point is actually given by the cross product of the two line l and l prime respectively here's a mathematical proof on uh how how this cross product works just now i have given you the geometric integration between behind why a point is uh represented as the cross product of two lines l1 and l prime now i'm going to give you a mathematical proof now suppose that we are given two lines l which is represented as a b c and a prime b prime and c the triplet scalar product identity uh gives this the triplet product uh scalar product identity it simply means that the dot products of the cross product of two lines are going to be zero which we can see that the dot pro the cross product actually uh of two vectors which you denote as l and l prime over here is going to be the line that is going to be the vector that we denote by v over here here this guy here is going to be l cross l prime over here and this is going to be octagonal to each other all these three lines they are all going to form they are going to be perpendicular to each other so what this means is that if we take the uh cross product of l and l prime we will get v which is octagonal to l and l prime and if we were to take the dot product of v with any one of these lines l or l prime over here since they're 90 degrees apart the the dot products of these two are going to be equals to 0 because dot products is simply if i have two lines and i take the dot product or two vectors i take the dot product the dot product is going to give me a b cosine theta equals to a dot with b the dot product of a and b is going to give me this so in the case where a and b are perpendicular to each other a a vector of a and b they are perpendicular to each other what this means is that the angle between them is going to be 90 degree hence the cosine of 90 degrees is going to give me a 0 over here hence this is going to give me a 0 of the relationship over here and if x is taught as the representing point then x lies on both lines l and l prime hence and hence it is the intersection of the two lines over here so we can see that uh because of this relation over here uh the triple scalar product identity over here this relation is going to give rise to zero so we can replace l and uh the cross product of l and l prime here with x over here in this the first case here and we can also see that this guy over here can also be denoted as x hence if we put this into the this this particular equation here we can see that from our earlier definition the incidence relation of x and l the point and a line when a point lies on a line it has to give a dot product of 0. this is arising from the equation of the line that we have seen earlier hence we can replace the cross product of l and l prime to be x where well we can conclude that the cross product of two lines is actually equals to the point itself and this is the algebraic derivation of the intersection of two lines that gives rise to a point and conversely we can also see that the cross product of two points x and x prime over here gives rise to a line l we can see this or we can observe this geometrically using this representation in the diagram over here so we have seen that two points is actually it can be represented as two rays in the homogeneous coordinates so for example x and x prime over here and the cross product of these two lines since they lie on the same plane it's going to give rise to a perpendicular uh vector a perpendicular normal vector over here which we denote as l over here and this l would represents this particular plane over here that intersects the plane k over here on at the line represented by l hence we say that the cross product of x and x prime uh is going to give rise to a line a vector that represents the line in the homogeneous coordinates form and algebraically the proof is given by this suppose that we are given two points x and x prime the triple product identity will use the same trick of the triple scalar product identity it gives x dot with the x cross x prime over here is going to be 0 as well as x prime dot product with x cross x prime is also going to be 0 which we can rewrite this here this part over here since the cross product is going to give another vector a three vector which we can write rewrite this as l and this part over here as l2 so and we can see that this is exactly the incidence relationship between a point and a line which we have mentioned earlier on and here we can simply write l to be equals to x cross x prime hence we can conclude that the cross product of two points in the homogeneous coordinates gives rise to a three vector line in the homogeneous coordinate and we can also see that there is no exception uh to be made of the intersection of two parallel lines so in cartesian coordinates this would become a problematic the intersection of two parallel lines because uh there is no way to represent this uh intersection in the cartesian coordinates as they simply do not intersect and and or they actually intersect at infinity so there is no way of representing infinity will end up to have a division by zero which would uh be not solvable in cartesian coordinates but in the homogeneous representation we can see that this becomes uh elegant to represent so now let's consider two parallel lines a x plus b y plus c equals to zero as well as uh a b uh a x plus b y plus c prime equals to zero which we represent in the homogeneous coordinate as a b c and a b c prime over here so we can see that since i mentioned that we can rewrite these equations into y equals to y equals to a over b minus or minus a b x minus c over b into the form of y equals to m x plus c so we can see that since a b in this tool are the same well and this means that in this case here they are the gradient which means that there are two parallel lines which simply intersects at the different point uh c and c prime respectively uh yeah and hence uh from a geometry or from a cartesian point of view we can easily see tell that if the two numbers of the lines are the same then uh these two lines are parallel to each other and the intersection of the two lines l and m prime which we have seen in the previous slide is given by the cross product of l and l prime and this can be evaluated in this particular form over here where we will simply get a scalar value of c prime minus c multiplied by the three vector that represents a point or homogeneous point in the projective space and in this case here the third number here is always going to be zero and ignore when we ignore the scale factor we can simply represent the point the intersecting point at b minus a and zero over here hence uh what this implies is that uh this is actually the intersection of the two parallel lines and it actually uh algebraically we can actually see that this point actually is the point at infinity which means that two parallel lines actually intersects at the point of infinity uh which can be represented very elegantly using homogeneous coordinates since the last value over here is zero when we convert this back into the cartesian coordinate we will get b divided by 0 minus a divided by 0 which is simply infinity and infinity that represents a point at infinity now example here is that consider two lines x equals to 1 and x equals to 2 over here so yeah in the geometrical way this is actually these are actually two lines uh parallel to each other at x equals to 1 and x equals to 2 or over here and these two lines are parallel and we will see that it consequently will intersect at infinity so using this cartesian representation over here there's no way of finding the intersection over here or representing it eloquently using mathematics but in homogeneous form we'll see that or we have already seen the derivation that this can be done in an elegant way so the two lines over here can be represented as l equals to minus 1 0 and 1 over here and the and as well as minus 1 0 and 2 over here which represents this particular two lines and and their intersection point can be uh given by the cross product of these two lines over here which is simply ends up to be zero one and zero we see the last coordinate is 0 this means that this point is actually a point at infinity in the y direction because 0 represents x and 1 represents y which means that the this vector over here this point here is pointing towards infinity and that's in the y direction which can be verified over here that since these two lines are parallel in the y direction they must intersect at infinity at the in the y direction so we have seen in the earlier example that any two parallel lines denoted by a b c and a b and c prime intersects at a point at infinity which is given by the ideal point of b minus a and 0 for all the c's and hence we also have seen that the ideal points will lie at the line at infinity so this is the line at infinity the ideal point is going to lie on this line at infinity and hence we can conclude that any two parallel line is always going to intersect at a ideal point at infinity which is going to be lying at the line at infinity hence any two lines parallelized is going to intersect the line at infinity at the ideal point and we have also seen that the ideal point is actually the direction of the is a it can be seen as a directional vector along the intersection of these two parallel lines in other words b minus a over here can be seen as the uh the tangent to the line which is the direction of the line that is pointing towards the point at infinity which is b and minus a over here and the octagonal to this line the tangent direction vector would be the normal of this line over here can be given by a and b over here and we can verify that these two are tangent to each other octagonal to each other uh b minus a and a b transpose over here by simply taking the dot products of these two guys or we can see that it will give us a b minus uh b a and that's going to be equals to zero the dot product of these two and we i've said earlier that the dot product of any two octagonal vectors a and b is going to end up to be zero which indeed verify that these two are the tangent to the line as well as the normal to the line and what's interesting here is that as the direction the line direction the parallel line direction varies a and b varies we can see that the ideal point that intersects at the line at infinity is going to also vary along the line at infinity since these two parallel lines are always going to intersect the line at infinity this mean but what it means here is that with the change of the direction of the lines the parallel lines the intersection at infinity is going to vary and this simply means that the tangent the direction of this line is parallelized is going to change with a and b hence the line infinity can be thought of a set of directions of the lines in the plane since all these are going to live in the 2d plane now through the dot products we can see or through the incidence relation of the that is denoted as the dot product we can see that the row of points and lines since there are all three vectors over here uh it can be interchangeable in the incidence equation in the dot product and the intersection of the two lines and the line through two points can also be uh interchangeable see all of the using the cross product over here so this leads to the observation of the duality principle which simply means that the point and the line they are dual to each other in the in terms of homogeneous coordinates representation and the operations in the protective space and the duality principle simply means that any theorem of two-dimensional projective space there exists a corresponding dual theorem that may be derived by interchanging the rows of the points and the lines in the original theorem so what this means here is that in the context of these guys over here what it means is that as long as we prove the cross product of two lines gives rise to a point there is no need for us to prove the the on the other hand where two points the cross product of two points will give us a line this is because due to the duality principle where line and points are interchangeable we simply just need to prove any one of this for the other one to hold true