sacar - Trazar el ángulo entre vectores

como sacar el angulo de un vector resultante

Question

Ya estoy buscando bastante tiempo para una función que traza el ángulo entre dos flechas / segmentos de línea. p.ej:

http://www.matrix44.net/cms/wp-content/uploads/2011/03/vector_dot_product.png

¿Se puede hacer esto fácilmente o tengo que encontrar la función de un segmento de un radio? Me sorprende que aún no haya encontrado nada, ya que R suele tener un paquete para todo.

Answer 1

Gracias bgoldst por tu esfuerzo. Me gustaría contribuir con una solución bastante sencilla que no necesariamente reproduzca la imagen anterior, pero que quizás sea útil para programadores menos experimentados como yo:

library(plotrix)

# Creates empty plot and variables
plot(1,type="n",axes=F,xlim=c(-0.1,1.2),ylim=c(-0.1,1.2),xlab="",ylab="")
x1 = c(1,0)
x2 = c(0.4,0.7)

# X1
arrows(0,0,x1[1],x1[2])
text(1,-0.05,expression(x[1]),cex=1.5)

# X2
arrows(0,0,x2[1],x2[2])
text(0.4,0.75,expression(x[2]),cex=1.5)


alpha_angle= acos((x2%*%x1)/(sqrt(x2%*%x2)*sqrt(x1%*%x1)))

draw.arc(0,0,0.15,angle2=alpha_angle)

text(0.07,0.05,expression(alpha),cex=1.5)

Al final, esto da la siguiente trama:

Answer 2

## helper variables and functions
tau <- pi*2;

circles <- function(x,y,r,fg,bg,lty,lwd,n.angle=2000,n.bg=2000,...) {
    comb <- cbind(x,y,r);
    angles <- seq(0,tau,len=n.angle);
    if (!missing(fg) && !is.null(fg)) fg <- rep(fg,len=nrow(comb));
    if (!missing(bg) && !is.null(bg)) bg <- rep(bg,len=nrow(comb));
    if (!missing(lty) && !is.null(lty)) lty <- rep(lty,len=nrow(comb));
    if (!missing(lwd) && !is.null(lwd)) lwd <- rep(lwd,len=nrow(comb));
    for (i in seq_len(nrow(comb))) {
        xc <- comb[i,''x''];
        yc <- comb[i,''y''];
        rc <- comb[i,''r''];
        xs <- xc+rc*cos(angles);
        ys <- yc+rc*sin(angles);
        ## optional bg
        if (!missing(bg) && !is.null(bg)) {
            bgc <- bg[i];
            rs.bg <- seq(r,-r,len=n.bg);
            xs.bg <- sqrt(rc^2 - rs.bg^2);
            ys.bg <- yc+rs.bg;
            segments(xc-xs.bg,ys.bg,xc+xs.bg,col=bgc,lty=1,lwd=1);
        }; ## end if
        args <- list(xs,ys);
        if (!missing(fg)) if (is.null(fg)) args[''col''] <- list(NULL) else args$col <- fg[i];
        if (!missing(lty)) if (is.null(lty)) args[''lty''] <- list(NULL) else args$lty <- lty[i];
        if (!missing(lwd)) if (is.null(lwd)) args[''lwd''] <- list(NULL) else args$lwd <- lwd[i];
        do.call(lines,c(args,...));
    }; ## end for
}; ## end circles()

radials <- function(x,y,a,r,...) {
    comb <- cbind(x,y,a,r);
    segments(comb[,''x''],comb[,''y''],comb[,''x'']+comb[,''r'']*cos(comb[,''a'']),comb[,''y'']+comb[,''r'']*sin(comb[,''a'']),...);
}; ## end radials()

circle.segments <- function(x,y,r,a1,a2,fg,bg,lty,lwd,n.angle=2000,fg.chord,fg.arc,...) {
    comb <- cbind(x,y,r,a1,a2);
    if (!missing(fg) && !is.null(fg)) fg <- rep(fg,len=nrow(comb));
    if (!missing(fg.chord) && !is.null(fg.chord)) fg.chord <- rep(fg.chord,len=nrow(comb)) else if (missing(fg.chord) && !missing(fg)) fg.chord <- fg;
    if (!missing(fg.arc) && !is.null(fg.arc)) fg.arc <- rep(fg.arc,len=nrow(comb)) else if (missing(fg.arc) && !missing(fg)) fg.arc <- fg;
    if (!missing(bg) && !is.null(bg)) bg <- rep(bg,len=nrow(comb));
    if (!missing(lty) && !is.null(lty)) lty <- rep(lty,len=nrow(comb));
    if (!missing(lwd) && !is.null(lwd)) lwd <- rep(lwd,len=nrow(comb));
    for (i in seq_len(nrow(comb))) {
        xc <- comb[i,''x''];
        yc <- comb[i,''y''];
        rc <- comb[i,''r''];
        a1c <- comb[i,''a1''];
        a2c <- comb[i,''a2''];
        angles <- seq(a1c,a2c,len=n.angle);
        tan.angles <- tan(angles);
        xs <- xc+rc*cos(angles);
        ys <- yc+rc*sin(angles);
        x1 <- xs[1];
        y1 <- ys[1];
        x2 <- xs[length(xs)];
        y2 <- ys[length(ys)];
        ## optional bg
        if (!missing(bg) && !is.null(bg)) {
            bgc <- bg[i];
            xs.chord <- seq(x1,x2,len=n.angle);
            ys.chord <- seq(y1,y2,len=n.angle);
            segments(xs.chord,ys.chord,xs,ys,col=bg,lty=1,lwd=1);
        }; ## end if
        ## chord segment
        args <- list(x1,y1,x2,y2);
        if (!missing(fg.chord)) if (is.null(fg.chord)) args[''col''] <- list(NULL) else args$col <- fg.chord[i];
        if (!missing(lty)) if (is.null(lty)) args[''lty''] <- list(NULL) else args$lty <- lty[i];
        if (!missing(lwd)) if (is.null(lwd)) args[''lwd''] <- list(NULL) else args$lwd <- lwd[i];
        do.call(segments,c(args,...));
        ## arc segment
        args <- list(xs,ys);
        if (!missing(fg.arc)) if (is.null(fg.arc)) args[''col''] <- list(NULL) else args$col <- fg.arc[i];
        if (!missing(lty)) if (is.null(lty)) args[''lty''] <- list(NULL) else args$lty <- lty[i];
        if (!missing(lwd)) if (is.null(lwd)) args[''lwd''] <- list(NULL) else args$lwd <- lwd[i];
        do.call(lines,c(args,...));
    }; ## end for
}; ## end circle.segments()

circle.sectors <- function(x,y,r,a1,a2,fg,bg,lty,lwd,n.angle=2000,fg.a1,fg.a2,fg.arc,...) {
    comb <- cbind(x,y,r,a1,a2);
    if (!missing(fg) && !is.null(fg)) fg <- rep(fg,len=nrow(comb));
    if (!missing(fg.a1) && !is.null(fg.a1)) fg.a1 <- rep(fg.a1,len=nrow(comb)) else if (missing(fg.a1) && !missing(fg)) fg.a1 <- fg;
    if (!missing(fg.a2) && !is.null(fg.a2)) fg.a2 <- rep(fg.a2,len=nrow(comb)) else if (missing(fg.a2) && !missing(fg)) fg.a2 <- fg;
    if (!missing(fg.arc) && !is.null(fg.arc)) fg.arc <- rep(fg.arc,len=nrow(comb)) else if (missing(fg.arc) && !missing(fg)) fg.arc <- fg;
    if (!missing(bg) && !is.null(bg)) bg <- rep(bg,len=nrow(comb));
    if (!missing(lty) && !is.null(lty)) lty <- rep(lty,len=nrow(comb));
    if (!missing(lwd) && !is.null(lwd)) lwd <- rep(lwd,len=nrow(comb));
    for (i in seq_len(nrow(comb))) {
        xc <- comb[i,''x''];
        yc <- comb[i,''y''];
        rc <- comb[i,''r''];
        a1c <- comb[i,''a1''];
        a2c <- comb[i,''a2''];
        angles <- seq(a1c,a2c,len=n.angle);
        xs <- xc+rc*cos(angles);
        ys <- yc+rc*sin(angles);
        ## optional bg
        if (!missing(bg) && !is.null(bg)) {
            bgc <- bg[i];
            segments(xc,yc,xs,ys,col=bgc,lty=1,lwd=1);
        }; ## end if
        ## a1 segment
        args <- list(xc,yc,xs[1],ys[1]);
        if (!missing(fg.a1)) if (is.null(fg.a1)) args[''col''] <- list(NULL) else args$col <- fg.a1[i];
        if (!missing(lty)) if (is.null(lty)) args[''lty''] <- list(NULL) else args$lty <- lty[i];
        if (!missing(lwd)) if (is.null(lwd)) args[''lwd''] <- list(NULL) else args$lwd <- lwd[i];
        do.call(segments,c(args,...));
        ## a2 segment
        args <- list(xc,yc,xs[length(xs)],ys[length(ys)]);
        if (!missing(fg.a2)) if (is.null(fg.a2)) args[''col''] <- list(NULL) else args$col <- fg.a2[i];
        if (!missing(lty)) if (is.null(lty)) args[''lty''] <- list(NULL) else args$lty <- lty[i];
        if (!missing(lwd)) if (is.null(lwd)) args[''lwd''] <- list(NULL) else args$lwd <- lwd[i];
        do.call(segments,c(args,...));
        ## arc segment
        args <- list(xs,ys);
        if (!missing(fg.arc)) if (is.null(fg.arc)) args[''col''] <- list(NULL) else args$col <- fg.arc[i];
        if (!missing(lty)) if (is.null(lty)) args[''lty''] <- list(NULL) else args$lty <- lty[i];
        if (!missing(lwd)) if (is.null(lwd)) args[''lwd''] <- list(NULL) else args$lwd <- lwd[i];
        do.call(lines,c(args,...));
    }; ## end for
}; ## end circle.sectors()

intersect.lines <- function(a1x,a1y,a2x,a2y,b1x,b1y,b2x,b2y) {
    comb <- cbind(a1x,b1x,a2x,b2x,a1y,b1y,a2y,b2y);
    comb <- array(comb,c(nrow(comb),2,2,2),dimnames=list(NULL,c(''a'',''b''),NULL,c(''x'',''y'')));
    any.points <- any(comb[,''a'',1,''x''] == comb[,''a'',2,''x''] & comb[,''a'',1,''y''] == comb[,''a'',2,''y'']) || any(comb[,''b'',1,''x''] == comb[,''b'',2,''x''] & comb[,''b'',1,''y''] == comb[,''b'',2,''y'']);
    any.points[is.na(any.points)] <- F;
    if (any.points) stop(''coincident 1 and 2 points.'');
    m.a <- (comb[,''a'',2,''y''] - comb[,''a'',1,''y''])/(comb[,''a'',2,''x''] - comb[,''a'',1,''x'']);
    m.b <- (comb[,''b'',2,''y''] - comb[,''b'',1,''y''])/(comb[,''b'',2,''x''] - comb[,''b'',1,''x'']);
    b.a <- comb[,''a'',1,''y''] - m.a*comb[,''a'',1,''x''];
    b.b <- comb[,''b'',1,''y''] - m.b*comb[,''b'',1,''x''];
    a.inf <- is.infinite(m.a);
    b.inf <- is.infinite(m.b);
    parallel <- ifelse(a.inf,ifelse(b.inf,T,F),ifelse(b.inf,F,m.a == m.b));
    x1equal <- comb[,''a'',1,''x''] == comb[,''b'',1,''x''];
    coincident <- ifelse(a.inf,ifelse(b.inf,x1equal,F),ifelse(b.inf,F,parallel & b.a == b.b));
    xi <- ifelse(coincident,Inf,ifelse(parallel,NaN,ifelse(a.inf,comb[,''a'',1,''x''],ifelse(b.inf,comb[,''b'',1,''x''],(b.b - b.a)/(m.a - m.b)))));
    yi <- ifelse(coincident,Inf,ifelse(parallel,NaN,ifelse(a.inf,m.b*comb[,''a'',1,''x''] + b.b,ifelse(b.inf,m.a*comb[,''b'',1,''x''] + b.a,m.a*xi + b.a))));
    xi[is.na(yi) & !is.nan(yi)] <- NA;
    yi[is.na(xi) & !is.nan(xi)] <- NA;
    cbind(x=xi,y=yi);
};

arrows.filled <- function(
    x1,y1,x2=x1,y2=y1,a=tau/32,al=a,ar=a,len=sqrt(diff(par(''usr'')[3:4])^2+diff(par(''usr'')[1:2])^2)/20,lenl=len,lenr=len,fg,bg,bgl,bgr,lty,lwd,
    fg.mainline,lty.mainline,lwd.mainline,
    fg.tipline,lty.tipline,lwd.tipline,
    fg.lwing,lty.lwing,lwd.lwing,
    fg.rwing,lty.rwing,lwd.rwing,
    fg.lcross,lty.lcross,lwd.lcross,
    fg.rcross,lty.rcross,lwd.rcross,
    ...
) {
    comb <- cbind(x1,y1,x2,y2,al,ar,lenl,lenr);
    if (!missing(fg) && !is.null(fg)) fg <- rep(fg,len=nrow(comb));
    if (!missing(lty) && !is.null(lty)) lty <- rep(lty,len=nrow(comb));
    if (!missing(lwd) && !is.null(lwd)) lwd <- rep(lwd,len=nrow(comb));
    if (!missing(fg.mainline) && !is.null(fg.mainline)) fg.mainline <- rep(fg.mainline,len=nrow(comb)) else if (missing(fg.mainline) && !missing(fg)) fg.mainline <- fg;
    if (!missing(fg.tipline) && !is.null(fg.tipline)) fg.tipline <- rep(fg.tipline,len=nrow(comb)) else if (missing(fg.tipline) && !missing(fg)) fg.tipline <- fg;
    if (!missing(fg.lwing) && !is.null(fg.lwing)) fg.lwing <- rep(fg.lwing,len=nrow(comb)) else if (missing(fg.lwing) && !missing(fg)) fg.lwing <- fg;
    if (!missing(fg.rwing) && !is.null(fg.rwing)) fg.rwing <- rep(fg.rwing,len=nrow(comb)) else if (missing(fg.rwing) && !missing(fg)) fg.rwing <- fg;
    if (!missing(fg.lcross) && !is.null(fg.lcross)) fg.lcross <- rep(fg.lcross,len=nrow(comb)) else if (missing(fg.lcross) && !missing(fg)) fg.lcross <- fg;
    if (!missing(fg.rcross) && !is.null(fg.rcross)) fg.rcross <- rep(fg.rcross,len=nrow(comb)) else if (missing(fg.rcross) && !missing(fg)) fg.rcross <- fg;
    if (!missing(lty.mainline) && !is.null(lty.mainline)) lty.mainline <- rep(lty.mainline,len=nrow(comb)) else if (missing(lty.mainline) && !missing(lty)) lty.mainline <- lty;
    if (!missing(lty.tipline) && !is.null(lty.tipline)) lty.tipline <- rep(lty.tipline,len=nrow(comb)) else if (missing(lty.tipline) && !missing(lty)) lty.tipline <- lty;
    if (!missing(lty.lwing) && !is.null(lty.lwing)) lty.lwing <- rep(lty.lwing,len=nrow(comb)) else if (missing(lty.lwing) && !missing(lty)) lty.lwing <- lty;
    if (!missing(lty.rwing) && !is.null(lty.rwing)) lty.rwing <- rep(lty.rwing,len=nrow(comb)) else if (missing(lty.rwing) && !missing(lty)) lty.rwing <- lty;
    if (!missing(lty.lcross) && !is.null(lty.lcross)) lty.lcross <- rep(lty.lcross,len=nrow(comb)) else if (missing(lty.lcross) && !missing(lty)) lty.lcross <- lty;
    if (!missing(lty.rcross) && !is.null(lty.rcross)) lty.rcross <- rep(lty.rcross,len=nrow(comb)) else if (missing(lty.rcross) && !missing(lty)) lty.rcross <- lty;
    if (!missing(lwd.mainline) && !is.null(lwd.mainline)) lwd.mainline <- rep(lwd.mainline,len=nrow(comb)) else if (missing(lwd.mainline) && !missing(lwd)) lwd.mainline <- lwd;
    if (!missing(lwd.tipline) && !is.null(lwd.tipline)) lwd.tipline <- rep(lwd.tipline,len=nrow(comb)) else if (missing(lwd.tipline) && !missing(lwd)) lwd.tipline <- lwd;
    if (!missing(lwd.lwing) && !is.null(lwd.lwing)) lwd.lwing <- rep(lwd.lwing,len=nrow(comb)) else if (missing(lwd.lwing) && !missing(lwd)) lwd.lwing <- lwd;
    if (!missing(lwd.rwing) && !is.null(lwd.rwing)) lwd.rwing <- rep(lwd.rwing,len=nrow(comb)) else if (missing(lwd.rwing) && !missing(lwd)) lwd.rwing <- lwd;
    if (!missing(lwd.lcross) && !is.null(lwd.lcross)) lwd.lcross <- rep(lwd.lcross,len=nrow(comb)) else if (missing(lwd.lcross) && !missing(lwd)) lwd.lcross <- lwd;
    if (!missing(lwd.rcross) && !is.null(lwd.rcross)) lwd.rcross <- rep(lwd.rcross,len=nrow(comb)) else if (missing(lwd.rcross) && !missing(lwd)) lwd.rcross <- lwd;
    if (!missing(bg) && !is.null(bg)) bg <- rep(bg,len=nrow(comb));
    if (!missing(bgl) && !is.null(bgl)) bgl <- rep(bgl,len=nrow(comb)) else if (missing(bgl) && !missing(bg)) bgl <- bg;
    if (!missing(bgr) && !is.null(bgr)) bgr <- rep(bgr,len=nrow(comb)) else if (missing(bgr) && !missing(bg)) bgr <- bg;
    for (i in seq_len(nrow(comb))) {
        x1c <- comb[i,''x1''];
        y1c <- comb[i,''y1''];
        x2c <- comb[i,''x2''];
        y2c <- comb[i,''y2''];
        alc <- comb[i,''al''];
        arc <- comb[i,''ar''];
        if (alc <= 0 || alc >= tau/2) stop(paste0(''arrow '',i,'' has invalid left angle '',alc,''.''));
        if (arc <= 0 || arc >= tau/2) stop(paste0(''arrow '',i,'' has invalid right angle '',alc,''.''));
        lenlc <- comb[i,''lenl''];
        lenrc <- comb[i,''lenr''];
        beta <- atan2(y2c-y1c,x2c-x1c);
        xl <- x2c - lenlc*cos(beta - alc);
        yl <- y2c - lenlc*sin(beta - alc);
        xr <- x2c - lenrc*sin(tau/4 - beta - arc);
        yr <- y2c - lenrc*cos(tau/4 - beta - arc);
        with(as.data.frame(intersect.lines(x1c,y1c,x2c,y2c,xl,yl,xr,yr)),{ e <- parent.env(environment()); e$xi <- x; e$yi <- y; });
        ## mainline
        args <- list(x1c,y1c,xi,yi);
        if (!missing(fg.mainline)) if (is.null(fg.mainline)) args[''col''] <- list(NULL) else args$col <- fg.mainline[i];
        if (!missing(lty.mainline)) if (is.null(lty.mainline)) args[''lty''] <- list(NULL) else args$lty <- lty.mainline[i];
        if (!missing(lwd.mainline)) if (is.null(lwd.mainline)) args[''lwd''] <- list(NULL) else args$lwd <- lwd.mainline[i];
        do.call(segments,c(args,...));
        ## bg left
        if (!missing(bgl) && !is.null(bgl)) {
            bglc <- bgl[i];
            polygon(c(x2c,xl,xi),c(y2c,yl,yi),border=NA,col=bglc);
        }; ## end if
        ## bg right
        if (!missing(bgr) && !is.null(bgr)) {
            bgrc <- bgr[i];
            polygon(c(x2c,xr,xi),c(y2c,yr,yi),border=NA,col=bgrc);
        }; ## end if
        ## tipline -- only draw if at least one tipline arg was given
        if (!missing(fg.tipline) || !missing(lty.tipline) || !missing(lwd.tipline)) {
            args <- list(xi,yi,x2c,y2c);
            if (!missing(fg.tipline)) if (is.null(fg.tipline)) args[''col''] <- list(NULL) else args$col <- fg.tipline[i];
            if (!missing(lty.tipline)) if (is.null(lty.tipline)) args[''lty''] <- list(NULL) else args$lty <- lty.tipline[i];
            if (!missing(lwd.tipline)) if (is.null(lwd.tipline)) args[''lwd''] <- list(NULL) else args$lwd <- lwd.tipline[i];
            do.call(segments,c(args,...));
        }; ## end if
        ## lwing
        args <- list(x2c,y2c,xl,yl);
        if (!missing(fg.lwing)) if (is.null(fg.lwing)) args[''col''] <- list(NULL) else args$col <- fg.lwing[i];
        if (!missing(lty.lwing)) if (is.null(lty.lwing)) args[''lty''] <- list(NULL) else args$lty <- lty.lwing[i];
        if (!missing(lwd.lwing)) if (is.null(lwd.lwing)) args[''lwd''] <- list(NULL) else args$lwd <- lwd.lwing[i];
        do.call(segments,c(args,...));
        ## rwing
        args <- list(x2c,y2c,xr,yr);
        if (!missing(fg.rwing)) if (is.null(fg.rwing)) args[''col''] <- list(NULL) else args$col <- fg.rwing[i];
        if (!missing(lty.rwing)) if (is.null(lty.rwing)) args[''lty''] <- list(NULL) else args$lty <- lty.rwing[i];
        if (!missing(lwd.rwing)) if (is.null(lwd.rwing)) args[''lwd''] <- list(NULL) else args$lwd <- lwd.rwing[i];
        do.call(segments,c(args,...));
        ## lcross
        args <- list(xl,yl,xi,yi);
        if (!missing(fg.lcross)) if (is.null(fg.lcross)) args[''col''] <- list(NULL) else args$col <- fg.lcross[i];
        if (!missing(lty.lcross)) if (is.null(lty.lcross)) args[''lty''] <- list(NULL) else args$lty <- lty.lcross[i];
        if (!missing(lwd.lcross)) if (is.null(lwd.lcross)) args[''lwd''] <- list(NULL) else args$lwd <- lwd.lcross[i];
        do.call(segments,c(args,...));
        ## rcross
        args <- list(xr,yr,xi,yi);
        if (!missing(fg.rcross)) if (is.null(fg.rcross)) args[''col''] <- list(NULL) else args$col <- fg.rcross[i];
        if (!missing(lty.rcross)) if (is.null(lty.rcross)) args[''lty''] <- list(NULL) else args$lty <- lty.rcross[i];
        if (!missing(lwd.rcross)) if (is.null(lwd.rcross)) args[''lwd''] <- list(NULL) else args$lwd <- lwd.rcross[i];
        do.call(segments,c(args,...));
    }; ## end for
}; ## end arrows.filled()

## basic plot outline
par(xaxs=''i'',yaxs=''i'');
xlim <- c(0,8);
ylim <- c(0,6);
extra <- 0.5;
plot(NA,xlim=xlim+extra*c(-1,1),ylim=ylim+extra*c(-1,1),axes=F,ann=F);

## custom axes
xtick <- 0:8;
ytick <- 0:6;
tick.len <- 0.06;
tick.zeroadd <- 0.1;
segments(xtick,0,xtick,-tick.len,lwd=2);
segments(0,ytick,-tick.len,ytick,lwd=2);
abline(h=0,lwd=2);
abline(v=0,lwd=2);
text(xtick[-1],-tick.len/2,xtick[-1],pos=1,font=2);
text(xtick[1]+tick.zeroadd,-tick.len/2,xtick[1],pos=1,font=2);
text(-tick.len/2,ytick[-1],ytick[-1],pos=2,font=2);
text(-tick.len/2,ytick[1]+tick.zeroadd,ytick[1],pos=2,font=2);

## define main points
V1 <- c(2,5);
V2 <- c(7,1);

## circle sector with label
V1.angle <- atan2(V1[2],V1[1]);
V2.angle <- atan2(V2[2],V2[1]);
sector.radius <- 2;
circle.sectors(0,0,sector.radius,V1.angle,V2.angle,''#277A27'',''#E5EFE5'',lwd=2);
label.radius <- 1.1;
label.angle <- mean(c(V1.angle,V2.angle));
text(label.radius*cos(label.angle),label.radius*sin(label.angle),''α'',family=''serif'',cex=1.3,col=''#277A27'');

## arrows
arrows.filled(0,0,V1[1],V1[2],a=tau*12/360,len=0.25,lwd=2,bg=''black'');
arrows.filled(0,0,V2[1],V2[2],a=tau*12/360,len=0.25,lwd=2,bg=''black'');

## point circles
circles(c(V1[1],V2[1]),c(V1[2],V2[2]),0.08,''black'',''blue'');

## point labels
text(V1[1]+0.03,V1[2]+0.2,sprintf(''V1 = (%d,%d)'',V1[1],V1[2]),pos=4,col=''blue'',font=2,family=''sans'');
text(V2[1]+0.05,V2[2]-0.2,sprintf(''V2 = (%d,%d)'',V2[1],V2[2]),pos=4,col=''blue'',font=2,family=''sans'');